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GRAPHS AND THEIRCOVERINGS

Jin Ho Kwak

Department of Mathematics, POSTECH, Pohang, 790–784 Korea

[email protected]

Roman Nedela

Matej Bel University, Banska Bystrica, Slovakia

Mathematical Institute, Slovak Academy of Sciences, Banska Bystrica, Slovakia

[email protected]

This work is supported by Com2MaC-KOSEF, Korea.

PrefaceCombinatorial treatment of graph coverings had its primary incentive inthe solution of Heawood’s Map Colour Problem due to Ringel, Youngsand others [Ri74]. That coverings underlie the techniques that led tothe eventual solution of the problem was recognized by Alpert andGross [GA74]. These ideas further crystalized in 1974 in the work ofGross [Gr74] where voltage graphs were introduced as a means of apurely combinatorial description of regular graph coverings. In paral-lel, the very same idea appeared in Biggs’ monograph [Bi74]. Much ofthe theory of combinatorial graph coverings in its own right was subse-quently developed by Gross and Tucker in the seventies. We refer thereader to [GT87, Wh84] and the references therein. The theory wasextended to combinatorial graph bundles introduced by Pisanski andVrabec [PV82] in the eighties.

After two decades, since the book of Gross and Tucker [GT87] waspublished, it remains the most popular textbook as well as the referencebook covering the topic. The authors of the presented monograph haveseveral ambitions.

Firstly, following the ideas in [MNS00] to establish and extend thecombinatorial theory of graph coverings and voltage assignments onto amore general class of graphs which include edges with free ends (calledsemiedges). The new concept of a graph proved to be useful in applica-tions as well as in theoretical considerations. Some of the applicationsare shown in the book. From the theoretical point of view, graphs withsemiedges arise as quotients of standard graphs by groups of automor-phisms which are semiregular on vertices and darts (arcs) but may fixedges. They may be viewed as 1-dimensional analogues of 2- and 3-dimensional orbifolds. The fruitful concept of an orbifold comes fromstudies of geometry of n-manifolds. Orbifolds and related concepts areimplicitly included in the work of pioneers such as Henri Poincare, orPaul Kobe. The first formal definition of an orbifold-like object was

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given by Ichiro Satake in 1956 in the context of Riemannian geome-try. William Thurston, later in the mid 1970s defined and named themore general notion of orbifold as part of his study of hyperbolic struc-tures. It is not a surprise that the concept of graphs with semiedgeswas discovered in context of theoretical physics, as well (see [GK98] forinstance).

Secondly, following the ideas in [GS93] and in the subsequent papers[MNS00, Ma98, Sir01, Ma02, MMP04, MP06] we develop and organizea material on the combinatorial treatment of the lifting automorphismproblem - a central problem considered in this book. The study ofthe automorphism lifting problem in the context of regular coverings ofgraphs had its main motivation in constructing infinite families of highlytransitive graphs. The first notable contribution along these lines ap-peared, incidentally, in 1974 in Biggs’ monograph [Bi74] and in a paperof Djokovic [Dj74]. Whereas Biggs gave a combinatorial sufficient con-dition for a lifted group to be a split extension, Djokovic found a crite-rion, in terms of the fundamental group, for a group of automorphismsto lift at all. A decade later, several different sources added furthermotivations for studying the lifting problem. These include: countingisomorphism classes of coverings and, more generally, graph bundles,as considered by Hofmeister [Ho91] and Kwak and Lee [KL90, KL92];constructions of regular maps on surfaces based on covering space tech-niques due to Archdeacon, Gvozdjak, Nedela, Richter, Siran, Skovieraand Surowski [ARSS94, AGJ97, GS93, NS96, NS97, Su88]; and con-struction of transitive graphs with prescribed degree of symmetry, forinstance by Du, Malnic, Nedela, Marusic, Scapellato, Seifter, Trofi-mov and Waller [DMW> 06, MM93, MMS> 06, MS93, ST97]. Liftingand/or projecting techniques play a prominent role also in the studyof imprimitive graphs, cf. Gardiner and Praeger [GP95] among others.The lifting problem in terms of voltages in the context of general topo-logical spaces was considered by Malnic [Ma98]. Venkatesh [Ve> 06]obtained structural results which refine the work of Biggs [Bi74, Bi84]and Djokovic [Dj74] on lifting groups. Recently, many original contribu-tions on the topic was published, both solving particular problems anddeveloping pieces of theory. However, there is no monograph covering(more or less completely) fundamentals of the theory. The topic is omit-ted in the book by Gross and Tucker and in other related monographs

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in topological graph theory. So far, the most complete contribution tothe theory was done in the thesis of A. Malnic [Ma96], some parts arecovered in the paper [MNS00].

The study of the lifting problem within the combinatorial frameworkwas of course preceded by that in the topological setting. We referthe reader to the work of Armstrong, Birman, Hilden, MacBeath andZieschang [Ar92, BH72, Mac61, Zi73] and many others.

Finally, it is our ambition to shift by one dimension up, namely, tostudy 2-cell embeddings of graphs into surfaces. The corresponding 2-dimensional objects are called maps and they present a central objectof topological graph theory. For the purpose of this future project,it was necessary first to rebuilt some classical parts of graph theoryand the presented monograph can be used as preliminary material forinvestigation of maps and their symmetries.

The book contains three chapters. First one introduce basic con-cepts such as graphs and groups acting on graphs. Some useful conceptsand statements from permutation groups are briefly mentioned. Thesecond chapter is central. There, a theory of graph coverings is devel-oped. It includes action of the fundamental group, classical approachto the theory of graph coverings and the associated theory of voltagespaces with some applications. The lifting automorphism problem isstudied in detail, theory of voltage spaces us unified and generalized tographs with semiedges. This way ‘a gap’ in the familiar and popularGross and Tucker’s book ‘Topological graph theory’ is filled in. In thefinal chapter the lifting and covering techniques are used to approachsome problems of classical graph theory. In particular, a material onflows on graphs and enumeration of graph coverings is included.

The book can be used as a material for a course on graph coveringsas well as it could be of interest of specialist interesting in the topic.

We would like to take this opportunity to thank all people who giveus valuable comments and suggestions for revising this book, includingmany students.

Finally we are grateful to Korea Science and Engineering Founda-tion (KOSEF) in Korea for supporting us during preparation of thisbook.

Jin Ho Kwak

iv

Roman NedelaE-mail: [email protected]

[email protected]

July 2005, in Pohang, Korea

Contents

Preface i

1 Graphs and Groups 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Graphs and their fundamental groups . . . . . . . . . . . 21.3 Group actions . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Vertex-transitive graphs . . . . . . . . . . . . . . . . . . 13

2 Coverings and liftings of automorphisms 232.1 Graph coverings and two group actions on the fibre . . . 232.2 Voltage spaces . . . . . . . . . . . . . . . . . . . . . . . . 302.3 Covering isomorphism problem . . . . . . . . . . . . . . 372.4 Lifting automorphism problem - classical approach . . . 422.5 Lifting of graph automorphisms in terms of voltages . . . 502.6 Lifting problem, case of abelian CT(p) . . . . . . . . . . 592.7 Lifting problem, case of elementary abelian CT(p) . . . . 69

3 Applications 813.1 Enumeration of Graph Coverings . . . . . . . . . . . . . 813.2 Some applications of graph coverings . . . . . . . . . . . 90

3.2.1 Existence of regular graphs with large girth . . . 913.2.2 Large graphs of given degree and diameter 2 . . . 913.2.3 Spectrum of a graph and its covering graphs . . . 93

3.3 Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Index 109

v

Chapter 1

Graphs and Groups

We begin with some basic definitions and elementary properties ofgraphs and coverings.

1.1 Introduction

In considerations in graph theory and more generally in combinatoricsone has to deal with a problem of representation and controlling prop-erties of large structures, potentially based on unbounded number ofelementary objects (vertices, points, edges). Several essentially differenttechniques to deal with large objects were developed. Most usual areinductive constructions and arguments employing inductive arguments,counting arguments which led to the birth and development of the prob-abilistic method in graph theory and finally, employing symmetries ofstructures one can use finite groups to construct highly symmetricalgraphs, designs, maps, geometries, etc, satisfying various combinato-rial properties. The latest approach is well-documented by a fruitfulconcept of a Cayley graph, introduced originally by group theorists in19-th century to investigate groups.

Topological graph theory deals with graphs as 1-dimensional topo-logical objects which are usually embedded to surfaces or to a morecomplex CW-complexes. The aim of this introductory chapter is todevelop a consistent combinatorial theory of such graphs. In partic-ular, epimorphisms between graphs which are locally bijective, calledgraph coverings, will be investigated. We demonstrate the usefulness of

1

2 Chapter 1. Graphs and Groups

graph coverings to solve several classical problems. Between the appli-cations the lifting automorphism problem is emphasized. There are tworeasons: firstly, the lifting automorphism problem and its applicationsin constructions of highly symmetrical graphs and maps received lastdecade a lot of attention in literature. Secondly, a purely combinatorialtreatment of this classical problem is not covered by the existing mono-graphs in topological graph theory, including White [Wh84], Gross andTucker [GT87].

Because of certain natural applications in the theory of Cayleygraphs as well as in the theory of maps on surfaces, and for the sakeof convenience, we extend our definition of a graph to allow multipleedges, loops and even semiedges. This slight modification has some far-reaching consequences. For instance, the fundamental group of such agraph may not be a free group(!). Therefore, it has been necessary tocarefully reexamine the whole basic theory of coverings as it is knownin topology and, so far, in topological graph theory. On the other hand,our revision has an additional pleasant consequence in that it enablesa smooth and unified transition to the theory of voltage spaces. Asa result, a genuine combinatorial theory of coverings with no directtopological counterpart is obtained.

The terminology not explicitly defined in this book (but freely usedin the sequel) is tacitly adopted from graph theory as well as grouptheory and topology [Ma91, Ro82, GT87].

1.2 Graphs and their fundamental groups

A graph is an ordered quadruple X = (D, V ; I, λ) where D is a set ofdarts, V is a nonempty set of vertices, which is required to be disjointfrom D, I is a mapping of D onto V , called the incidence function, and λis an involutory permutation of D, called the dart-reversing involution.For convenience or if λ is not explicitly specified we sometimes writex−1 instead of λx. Intuitively, the mapping I assigns to each dartits initial vertex, and the permutation λ interchanges a dart and itsreverse. The terminal vertex of a dart x is the initial vertex of λx. Theset of all darts initiated at a given vertex u is denoted by Du, calledthe neighborhood of u. The cardinality |Du| of Du is the valency of thevertex u. The orbits of λ are called edges; thus each dart determines

1.2. Graphs and fundamental groups 3

uniquely its underlying edge. An edge is called a semiedge if λx = x,a loop if λx 6= x and Iλx = Ix, and it is called a link otherwise. Werepresent a graph, as defined above, by a topological space in the usualway as a 1-dimensional CW-complex. Note that from a topologicalpoint of view a semiedge is identical with a pendant edge except thatits free endpoint is not listed as a vertex.

There is an alternative definition of our graph discovered in contextof theoretical physics (see [GK98]). They introduce a graph as a triple(D;∼, λ), where D is the set of darts, ∼ is an equivalence relation onD and λ is an involutory permutation acting on D. The equivalencerelation ∼ gives rise to a decomposition of D into equivalence classes[x], x ∈ D. We may set V = [x] | x ∈ D as the quotient set andI : x → [x]. On the other hand, for a given graph X = (D, V ; I, λ)the corresponding equivalence relation is defined by the decompositionD = ∪v∈V I−1(v). Thus the two definitions are equivalent.

Our definition of a graph is more convenient for purposes of thisbook than the classical definition of a graph consisting of vertices Vand edges E ⊆ 2V of cardinality two, since in topological graph theorymultiple edges, loops or semiedges naturally arise. In what follows ourinvestigation is restricted to unoriented graphs. However, in some casesan orientation of edges is used as a technical tool. An orientation of agraph X = (D, V ; I, λ) is a transversal D+ of the set of edges E(X) =x, x−1 | x ∈ D. It follows that D+ intersects each edge in exactlyone dart. With each orientation D+ a unique (reverse) orientation D−

such that D+ ∪ D− = D is associated. The intersection D+ ∩ D− isformed by the darts underlying the semiedges of X and it is emptyfor graphs without semiedges. An oriented or directed graph is a graphwith a specified orientation.

A morphism of graphs f : (D,V ; I, λ) → (D′, V ′; I ′, λ′) is a functionf : D ∪ V → D′ ∪ V ′ such that fD ⊆ D′, fV ⊆ V ′, fI = I ′f andfλ = λ′f . Thus, a morphism is an incidence-preserving mapping whichtakes vertices to vertices and edges to edges. Note that the image of alink can be a link, a loop or a semiedge, the image of a loop can be aloop or a semiedge, and the image of a semiedge can be just a semiedge.Composition of morphisms is defined as composition of functions fromright to left. This defines the category Grph. Note that the condition


fI = I ′f implies that f is completely determined by its restriction f |Don the set of darts. Moreover, if X is a simple graph (one withoutloops, semiedges or multiple adjacencies) then f |V coincides with theusual notion of a graph morphism used commonly by graph theorists.

The isomorphism between two graphs and the automorphism (group)of a graph are defined in a natural way. The automorphism group of agraph X will be denoted by Aut(X).

A walk of length n ≥ 1 is a sequence of n darts W = x1x2 . . . xn suchthat, for each index 1 ≤ k ≤ n− 1, the terminal vertex of xk coincideswith the initial vertex of xk+1. Moreover, we define each vertex to bea trivial walk of length 0. The initial vertex of W is the initial vertexof x1, and the terminal vertex of W is the terminal vertex of xn. Thewalk is closed if the initial and the terminal vertex coincide. In this casewe say that the walk is based at that vertex. If W has initial vertexu and terminal vertex v, then we usually write W : u → v. The walkW−1 = x−1

n x−1n−1 . . . x−1

1 is the inverse of W . Let W1 and W2 be twowalks such that the terminal vertex of W1 coincides with the initialvertex of W2. We define the product W1W2 as the juxtaposition of thetwo sequences. The product of a walk W and a trivial walk is W itself.A graph is connected if for every pair of vertices u and v there exists awalk u → v. A walk W is reduced if it contains no subsequence of theform xx−1. Clearly, each walk gives rise to a unique reduced walk by arepeated cancellation of all occurrences of the form xx−1. For instance,if x is a dart underlying a semiedge, then xx is a closed walk whichreduces to the trivial walk at the initial vertex of x. By declaring twowalks to be (combinatorially) homotopic provided that they give riseto the same reduced walk, we obtain an equivalence relation similar tothe usual homotopy relation known in topology. The essential differencefrom the homotopy on the associated 1-CW complex is that a walk oflength 1 traversing a semiedge is not homotopically trivial.

By π(X) we denote the fundamental groupoid of a graph X, that is,the set of all reduced walks equipped with the product W1 ·W2 (oftendenoted simply by W1W2) being the reduction of W1W2, whenever de-fined. Note that, for every vertex u, the trivial walk 1u behaves as thelocal identity in the groupoid.

Lemma 1.1. The subset π(X, u) ⊆ π(X) of all reduced closed walks

1.2. Graphs and fundamental groups 5

based at a vertex u ∈ X forms a group. If X is connected, then thegroups π(X, u), π(X, v) at any two vertices u, v are isomorphic.

The group πu(X) := π(X, u) is called the fundamental group of Xat u. Sometimes it is denoted simply by πu.

Proof. We prove the second part. Since X is connected, any two ver-tices u, v are connected by a walk W : u → v. Then the mappingS 7→ W−1SW takes a u-based closed walk S ∈ π(X, u) onto a closedv-based walk in π(X, v). It is easily seen that it preserves the ho-motopy classes, composition of classes and the identity is mapped tothe identity. Moreover, if S1, S2 ∈ πu then W−1S1W = W−1S2W im-plies S1 = S2. Hence S 7→ W−1SW gives a monomorphism πu → πv.Clearly, its inverse is S 7→ WSW−1. Thus πu ∼= πv. 2

In contrast to the classical case of graphs without semiedges, thefundamental group need not be a free group in our case. Nevertheless,it is a free product of cyclic groups, each of which is isomorphic to Zor Z2.

u u

I I

k¸

I

±

¼N

)W+

Q

x- x1

¾

-

?

xk-

Qk

P

Xk Yk

(a) A fundamental cycle (b) Factoring into fundamental cycles

Figure 1.1: A generator of πu


Lemma 1.2. Let X be a finite connected graph with e edges, s ≤ esemiedges and v vertices. Then the fundamental group π(X, u) is a freeproduct of e − v + 1 cyclic groups. More precisely, it is a free productof e− s− v + 1 copies of Z and s copies Z2.

Proof. A minimal generating set of π(X, u) can be constructed by thestandard algorithm employing a spanning tree T of X (an inclusion-minimal connected spanning subgraph) as in the classical case. Letx, λx be a cotree edge and P , Q be the unique u → Ix, u → Ix−1

paths in T . We form a generating set from the homotopy classes[PxQ−1], where x ranges through the cotree darts choosing exactlyone of x, λx for each cotree edge. See Figure 1.1(a). Such u-basedclosed walk PxQ−1 is called the fundamental cycle associated with thecotree dart x. The proof is finished by showing that each homotopyclass of u-based closed walks can be expressed in terms of generators.We first reduce a u-based closed walk to a minimal representative Wof the homotopy class in πu. Let W = · · ·x1 · · ·x2 · · · xk · · · , wherex1, x2, . . . , xk are the cotree darts, or their inverses, listed in the orderas they appear in W . Let Wi be the subwalk joining xi to xi+1. By theminimality, Wi is a path for all i = 1, 2, . . . , k−1 and Wk = YkQ

−1k QkXk

is a concatenation of four paths as in Figure 1.1(b), where Qk belongsto T and joins u to the path Wk = YkXk in T , the terminal vertex of Yk

and Qk coincides. Let Qi be the unique path in T joining u to Wi in theshortest length. The terminal vertex of Qi is met by Wi exactly once.Thus Wi = YiXi where Yi and Xi are, respectively, the subpaths termi-nating and initiating at the terminal vertex of Qi. Then replacing eachWi by YiQ

−1i QiXi we do not change the homotopy class of W . This

shows that W ∼ QkXkx1Y1Q−11

∏k−1i=1 QiXixi+1Yi+1Q

−1i+1 expressing W

as a product of the generating walks. Clearly, no generating homotopyclass can be expressed as a product of the others. Hence the generatingset is a minimal one.

Moreover, since the order of the class [PxQ−1] is two if and onlyif x is a semiedge (in that case P = Q), the above algorithm yields aone-to-one correspondence between the set of semiedges of X and theset of free factors in π(X, u) isomorphic to Z2. 2

The number β(X) = e− v + 1 is called the betti number of a graphX.

Group actions 7

Remark: By the abelianization of a group G we mean a factor groupH = G/[G,G], where [G,G] is the derived group. Abelianization ofthe fundamental group π(X, u) is called the (first) homology group ofa graph X and denoted by H1(X). By Lemma 1.1 H1(X) does notdepend on the choice of the base vertex. By Lemma 1.2 the (first)homology group H1(X) of a graph X is isomorphic to the free productof the free abelian group of rank e− s− v + 1 and s copies of Z2.

Exercises

1.2.1. Describe connected graphs with abelian fundamental groups.1.2.2. Prove that each class of homotopic equivalent walks contains a unique

reduced walk.1.2.3. Let X be a finite connected graph and T ≤ X be a spanning tree.

Prove that for any two distinct cotree darts xi+1, xj+1, the u-basedclosed walks QiXixi+1Yi+1Q

−1i+1 and QjXjxj+1Yj+1Q

−1j+1, defined in

the proof of Lemma 1.2, are not homotopic equivalent.

1.3 Group actions

Graph coverings and group actions are closely related. It is thereforeimpossible to develop a consistent theory of graph coverings and notto refer to some basic statements from the theory of group actions. Inwhat follows we introduce basic notions and statements without proofs.For a more detailed explanation the reader is referred to [DM96, Wie].

Let Ω be a set. By SymL Ω and SymR Ω we denote the left and theright symmetric group on a (nonempty) set Ω, respectively. The rightsymmetric group on the set 0, 1, . . . , n − 1 is commonly denoted bySn.

A (right) action of a group G on a set Ω is defined by a functionσ : Ω × G → Ω, with the common notation σ(z, g) = z · g, such thatz ·1G = z and z · (gh) = (z ·g) ·h for any z ∈ Ω and g, h ∈ G. Note thatthe function z 7→ z · g is a bijection on Ω for each g in G. Therefore,one may equivalently define a group action of G on a set Ω as a grouphomomorphism G → SymR Ω. When a group G acts on a set Ω, we callΩ a G-space.

The left action is defined in a similar manner, as a function G×Ω →Ω, or a group homomorphism G → SymL Ω.


The subgroup Gz = g ∈ G | z · g = z is called the stabilizer ofz ∈ Ω under the action of G. The set z · g ∈ Ω | g ∈ G is called theorbit of z. An action of a group is called semi-regular if all the stabilizersare trivial. It is transitive if it has only one orbit, that means for anytwo z2, z1 ∈ Ω there is g ∈ G such that z2 = z1 · g. The stabilizersin a transitive action are all conjugate, thus the structure of Gz doesnot depend on the choice of the point z. A transitive and semi-regularaction is called regular.

Let G act on Ω. The core CoreΩ(G) = ∩z∈ΩGz ≤ G is formed bythe elements of G letting all the elements fixed.

If a G-action on Ω is defined by a homomorphism α : G → SΩ,then the core CoreΩ(G) is nothing but the kernel of α. Hence, the coreis a normal subgroup of G. If CoreΩ(G) = 1 then the action of G onΩ is faithful. If G acts faithfully on Ω then the assignment g 7→ αg,where αg(x) = x · g, is a monomorphism G → SΩ, called a permutationrepresentation of G on Ω and one can identify G with a subgroup ofSΩ. In a general case, G/CoreΩ(G) is isomorphic to a subgroup of SΩ.The size |Ω| is called a degree of the permutation representation. Agroup of permutations G is of degree n if G ≤ SΩ and |Ω| = n.

We shall later need the concept of a group action extended to an ac-tion of a fundamental groupoid which can be introduced in the obviousway, see Proposition 2.2.

Let Ωi be a Gi-space for i = 1, 2. We call a group isomorphismf : G1 → G2 admissible or consistent with the respective actions ifthere exists a bijection φ : Ω1 → Ω2 such that φ(z · g) = φ(z) · f(g) forall z ∈ Ω and g ∈ G, that is, the diagram

Ω1 ×G1 Ω1

Ω2 ×G2 Ω2

-

-?

φ× f?φ

commutes. For a Gi-space Ωi (i = 1, 2), if there is an admissible iso-morphism f : G1 → G2, then two actions are called isomorphic and thepair (φ, f) is called an isomorphism of the actions.

1.3. Group actions 9

As a special case, let G1 = G2 := G, and f = idG be the identity.We define a morphism from Ω1 to Ω2 to be a function φ : Ω1 → Ω2

such that φ(z · g) = φ(z) · g for all g ∈ G and all z ∈ Ω1. Morphismsof G-spaces are also called equivariant. If an equivariant morphismφ : Ω1 → Ω2 is bijective then its inverse is also equivariant, and φ isan equivalence or G-isomorphism. The two G-spaces Ω1 and Ω2 areisomorphic.

Transitive permutation representations are of crucial importance forinvestigation of highly transitive combinatorial structures. The follow-ing two statements are well-known, see [DiMo, Robinson p.35].

Let H ≤ G be a subgroup. For each g ∈ G define τg by τg :Hx 7→ Hxg. Then τ defines a transitive action of G on the right cosets(by right translation) with kernel HG =

⋂x∈A

g−1Hg. It follows that

p : g 7→ τg is a transitive permutation representation of the quotientgroup G/HG. In particular, if H = 1 then the action is regular, andthe statement is known as a Cayley Theorem.

The next theorem shows that all transitive actions can be obtainedas above up to equivalence.

Theorem 1.3. Any transitive action of a finite group G on a set Ω isequivalent to the action of G on the right cosets of some subgroup H ofG by right translation.

Since the kernel of the action of G on the right cosets of H is thecore CoreG(H) =

⋂g∈G g−1Hg for some H ≤ G, it follows that for

an abelian group G, the right translation of G is the only (faithful)transitive representation of G up to equivalence.

Definition 1.1. Let G ≤ SΩ, ∆ ⊆ Ω. If for any g ∈ G, either ∆g = ∆or ∆g ∩∆ = ∅, then we call ∆ a block of G.

Obviously, Ω, ∅ and one-element subset α are blocks of G; theyare called the trivial blocks.

Definition 1.2. Let G ≤ SΩ. If G has a nontrivial block ∆, then Gis called an imprimitive group, and ∆ is called an imprimitivity set.Otherwise, we say G a primitive group. When |Ω| = 2, we make thefollowing convention: if G is identity, we also call G imprimitive.


In other words, a permutation group G on a set Ω is imprimitiveif Ω may be split into nontrivial blocks (of the same size if the actionis transitive) such that each g ∈ G maps a block into a block. Forexample, all intransitive groups are imprimitive.

Obviously, every primitive group is transitive. Until the end of thesection, we assume that G is a transitive permutation group on Ω.

Proposition 1.4. Let G ≤ SΩ. Assume that G is transitive but notprimitive. Let ∆ be an imprimitivity set of G. Write H = G∆ = g ∈G | ∆g = ∆. Then

(1) the subgroup H is transitive on ∆.

(2) Assume that G =⋃

r∈R Hr is a decomposition of G into a unionof right cosets of H. Then Ω =

⋃r∈R ∆r, and for any r 6= r′ in

R, we have ∆r ∩∆r′ = ∅.(3) |∆| divides |Ω|.

The set ∆r | r ∈ R is called a complete system of imprimitivity.

Proof. (1) For any α, β ∈ ∆, since G is transitive, there is a g ∈ G suchthat αg = β. Thus β ∈ ∆ ∩∆g, and ∆ ∩∆g 6= ∅. Since ∆ is a block,we get ∆ = ∆g. Hence g ∈ H, and H is transitive on ∆.

(2) Assume that G =⋃

r∈R Hr is a decomposition of G into a unionof right cosets of H. Let α ∈ ∆, β ∈ Ω. Since G is transitive, there is ag ∈ G such that β = αg ∈ ∆g. Let g = hr, where h ∈ H, r ∈ R. Thenwe have β ∈ ∆hr = ∆r and Ω =

⋃r∈R ∆r.

Moreover, if ∆r ∩ ∆r′ 6= ∅ then ∆rr′−1 ∩ ∆ 6= ∅. Since ∆ is aimprimitivity set, we have ∆ = ∆rr′−1

. Hence rr′−1 ∈ H, Hr = Hr′.Since R is a complete set of representatives of H, we have r = r′.

(3) Since |∆| = |∆r|, the conclusion follows from (2). 2

Corollary 1.5. A transitive group of prime degree is primitive.

The kernel K of an action of G on Ω is a subgroup formed byelements of G fixing each block of imprimitivity in ∆.

Proposition 1.6. The kernel K of an action of G with respect toimprimitivity system ∆ is a normal subgroup of G

1.3. Group actions 11

The following theorem is well-known in a permutation group theoryand will be used later.

Theorem 1.7. (see [DM96, Theorem 4.2A]) Let G be a transitive sub-group of Sym(Ω), and let C be the centralizer of G in Sym(Ω). Then

(1) C is semiregular and C ∼= NG(Gx)/Gx for any x ∈ Ω.

(2) C is transitive if and only if G is regular,

(3) if C is transitive then it is conjugate to G in Sym(Ω) and henceC is regular.

So far we have considered action of a group on its subgroups by leftand right translations. There is another important action of a groupon its subgroups, namely the action by conjugation. If H < G is asubgroup then the stabilizer NG(H) = g ∈ G|H = Hg = g−1Hg iscalled the normalizer of H in G. Clearly, the normalizer is a normalsubgroup of G containing H. Moreover, it is the least subgroup ofH with this property. Orbits of the action of G are called conjugacyclasses. Two groups H and Hg in the same conjugacy class are calledconjugate subgroups. Counting conjugacy classes of subgroups of finiteindex in a finitely generated group is closely related to enumeration ofisomorphism classes of combinatorial objects. To this end we presentthe following counting lemma recently proved by A. Mednykh [Me06].

Theorem 1.8. Let G be a finitely generated group. Then the numberof conjugacy classes of subgroups of index n in the group G is given bythe formula

NG(n) =1

n

∑

`|n` m=n

∑K<mG

Epi(K, Z`),

where the sum∑

K <mG is taken over all subgroups K of index m in thegroup G and Epi(K, Z`) is the number of epimorphisms of the groupK onto a cyclic group Z` of order `.

We show how to compute the number of homomorphisms and epi-morphisms of an arbitrary finitely generated group into cyclic Z`. De-note by Hom(G, Z`) the set of homomorphisms of a group G into thecyclic group Z` of order `. Since |Hom(G, Z`)| =

∑d| `|Epi(G, Zd)|, by

the Mobius inversion formula we have the following result


Lemma 1.9. (G. Jones [Jo95])

|Epi(G, Z`)| =∑

d| `µ(

`

d)|Hom(G, Zd)|,

where µ(n) is the Mobius function.

This lemma essentially simplifies the calculation of|Epi(G, Z`)| for a finitely generated group G. Indeed, let H1(G) =G/[G, G] be the first homology group of G. Suppose that H1(G) =Zm1 ⊕ Zm2 ⊕ . . .⊕ Zms ⊕ Zr. Then we have

Lemma 1.10.

|Epi(G, Z`)| =∑

d| `µ(

`

d) (m1, d) (m2, d) . . . (ms, d) dr,

where (m, d) is the greatest common divisor of m and d.

Note that |Hom(Zm, Zd)| = (m, d) and |Hom(Z, Zd)| = d. Since thegroup Zd is Abelian, we obtain

|Hom(G, Zd)| = |Hom(H1(G), Zd)|= (m1, d) (m2, d) . . . (ms, d) dr.

Then the result follows from Lemma 1.9.

Corollary 1.11. Let Fr be a free group of rank r. Then H1(Fr) = Zr

and Epi(Fr,Z`) =∑d| `

µ( `d)dr.

Exercises

1.3.1. Let P1 : Hg 7→ Hgx and P2 : Kg 7→ Kgx be two actions of a group Gon right cosets of subgroups H ≤ G and K ≤ G, respectively. Showthat P1 and P2 are equivalent if and only if H and K are conjugate inG.

1.3.2. (Orbit-Stabilizer Theorem) Prove: Let G act on a set Ω. Thenfor each z ∈ Ω, |G/Gz| equals to the size of the orbit of z.

Vertex-transitive graphs 13

1.3.3. (Burnside Lemma) Prove: Let G act on a set Ω. For g ∈ G, theset of fixed points of g is

fixΩ(g) = x ∈ Ω∣∣ x · g = x.

Then the number of orbits is the average of the number of fixed points,i.e., |Ω/G| = 1

|G|∑

g∈G |fixΩ(g)|. In particular, if G is transitive on Ω,then |G| = ∑

g∈G |fixΩ(g)|; and when |Ω| > 1, G has a regular element,that is an element without fixed points.

1.4 Vertex-transitive graphs

Group actions can be used to construct highly symmetrical graphs. Onecan use them to illustrate the notions and statements defined above.As we shall see later, the construction of a Cayley graph naturally gen-eralize to Cayley voltage graphs and graphs presented here will providea wide class of useful examples.

Cayley graphs. Given a group G and a set of generators S = S−1,1 /∈ S, we define a Cayley graph Cay(G,S) = (D, V ; I, λ) by settingD = G×S, V = G, I(g, x) = g and λ(g, x) = (gx, x−1). Clearly, G actson V by left multiplication g · v = gv. Let τg denotes the permutationgiven by the left multiplication by g. It is easy to check τg is a graphautomorphism, and the set A = τg | g ∈ G with the compositionof permutations form a subgroup of Aut(Cay(G,S)) isomorphic to G.Since A acts transitively (even regularly) on V , Cayley graphs form afamily of highly symmetrical examples of graphs. Note that G also actstransitively (even regularly) on V by right multiplication, v · g = vg,but it may not be a graph automorphism. Although A ∼= G the leftand right actions are not isomorphic, in general.

There is a natural coloring f of edges of a Cayley graph Cay(G, S) bythe pairs of generators of the form x, x−1 defined by f(gh) = x, x−1if g−1h ∈ x, x−1. The monochromatic factor determined by x, x−1is either a 2-factor formed by |G|/|x| cycles of length |x| > 2, or |x| = 2and it is a perfect matching. It is easy to show that the left action ofG on vertices preserves the coloring of edges.

The two actions GL ≤ SG, GR ≤ SG of G by left and right trans-lation on itself can be viewed as a particular instance of Theorem 1.7


with GRx = 1 and C = GL ≤ Aut(Cay(G,S)).The following characterization of Cayley graphs is well-known.

Proposition 1.12. (Sabidussi) A simple graph X is a Cayley graph ifand only if its automorphism contains a subgroup H ≤ Aut(X) actingregularly on vertices.

Proof. Let X = Cay(G,S) be a Cayley graph. It was already observedthat H = τg | g ∈ G ≤ Aut(X) is a transitive group of automor-phisms. Indeed, if g and h are two vertices of Cay(G,S) then there isan x ∈ G such that h = xg = τx(g) proving the transitivity. Assumingτgx = x for some vertex x ∈ G, we get gx = x implying g = 1. Hence,Gx = τ1 is trivial and the action of H is regular.

For the sufficiency, let H ≤ Aut(X) be a subgroup acting regularlyon the vertices of X. Fix a vertex w. Since the action is regular, foreach vertex u there is a unique element g ∈ H taking w onto u. Itgives a bijective labelling ` : V → H such that u = `(u)w for eachvertex u and we may identify the vertices of X with elements of H. Letuv be an arbitrary edge, and let x = `(u) takes an edge wa onto uv.Then u = xw and v = xa. Let g = `(a). By taking labellings in H,we can say that x = `(u) takes an arc (1, g) to an arc (x, xg) which isthe label of (u, v). Hence the adjacency of vertices is defined by rightmultiplication by the labels of the neighbours of w with `(w) = 1. SetS to be the set of labels of the neighbours of w. If g = `(a) belongsto S then g−1 takes a 7→ w and w 7→ b, where g−1 = `(b) ∈ S. HenceS = S−1 is closed under taking inverses and we get X = Cay(G,S). 2

Note that a representation of a graph X as a Cayley graph may notbe unique. For instance, the complete graph Kn can be defined as aCayley graph Cay(G; G \ 1) for any group of order n.

Example 1.1. (Octahedral graphs) The octahedral graph is definedby

On = K2n − k, n + k | k = 1, . . . , n = K2n − n disjoint edges.

For example, O2 is the cycle C4 and O3 is the underlying graph of theregular octahedron. The octahedral graph can be defined as a Cayley

1.4. Vertex-transitive graphs 15

graph On = Cay(Z2n; ±1,±2, . . . ,±(n − 1)) as well. However, the3-dimensional octahedral graph O3 can be defined alternatively as

O3 = Cay(S3; (1, 2, 3), (1, 3, 2), (1, 2), (2, 3)).

Both representations of O3 are depicted on Fig. 1.3 with r = (1 2 3)and s = (1 2) in S3.

1

2

3

4

1

2

3

4

5

6

1

2

34

5

6

7 8

O2 O3 O4

Figure 1.2: Octahedral graphs On

..............................

............................

....................

..............................................

........

........

..................................

0

1

2

3

45

..............................

.....................................................................................

................................................

..........

1

s

r

sr−1

r2

sr

(a) Cay(Z6;±1,±2) (b) Cay(S3;±r,±s)

Figure 1.3: Two presentations of the octahedral graph O3


Example 1.2. (Generalized Petersen graphs) All Cayley graphsare vertex-transitive but the converse is not true. A simple family ofvertex-transitive graphs that are not Cayley graphs can be constructedas follows. A generalized Petersen graph GP (n, k), k ≤ n/2 is a simplegraph whose vertices are elements of Zn×Z2 and each vertex (i, j) hasexactly three neighbours (i + kj, j), (i− kj, j) and (i, j + 1). It followsthat a mapping ρ taking (i, j) 7→ (i + 1, j) is a graph automorphism.Hence, either Aut(GP (n, k)) acts on the vertex set with two orbitsZn×0, Zn×1; or it is vertex-transitive. If k2 ≡ ±1(mod n) there isan additional automorphism α of GP (n, k) taking α : (i, j) 7→ (ki, j+1).This can be easily checked by computing images of the three kinds ofpairs of adjacent vertices. Hence the graph is vertex-transitive in thiscase. The goal is that with a single exception the opposite implicationholds as well.

The following statements classify those GP (n, k) that are vertex-transitive or Cayley graphs.

Theorem 1.13. (Frucht, Graver, Watkins [FGW71]) A generalized Pe-tersen graph GP (n, k) is vertex-transitive if and only if either (n, k) =(10, 2), or k2 ≡ ±1(mod n).

Theorem 1.14. [NS95] A generalized Petersen graph GP (n, k) is aCayley graph if and only if k2 ≡ 1(mod n).

Proof. We prove only the sufficiency. Assume k2 ≡ 1(mod n). Takethe subgroup H = 〈ρ, α〉 ≤ Aut(GP (n, k)). It follows that

αρα(i, j) = αρ(ki, j + 1)

= α(ki + 1, j + 1) = (k(ki + 1), j) = (i + k, j) = ρk(i, j).

Hence, we have αρα = ρk, 〈ρ〉 is a normal subgroup of H of index 2and so |H| = 2n. Therefore, the action of H is regular on vertices. 2

In view of Theorem 1.3 we generalize the construction of a Cayleygraph as follows.

Coset graphs. Let G be a finite group and H a proper subgroup ofG with ∩g∈GHg = 1. By a double coset we mean a set of elements


of the form HxH, where x ∈ G. Let B be a union of some dou-ble cosets of H in G such that B = B−1 and G = 〈H, B〉. Denoteby X = C(G; H, B) the coset graph with V (X) = gH

∣∣ g ∈ G,D(X) = (gH, gbH)

∣∣ b ∈ B, g ∈ G, I(gH, gbH) = gH and thedart-reversing involution L(gH, gbH) = (gbH, gbb−1H) = (gbH, gH).Clearly, G acts faithfully and vertex-transitively on the coset graphby left multiplication as group of graph automorphisms with a vertexstablizer isomorphic to H.

Conversely, let X = (D,V ; I, λ) be a simple vertex-transitive graphand let A ≤ Aut(X) acts vertex-transitively. Given a fixed vertexv in X, let H := Av be the stabilizer. Then the left cosets A/Hcan be identified with the vertex set V by correspondence gH 7→vg. Let d1, d2, . . . , d` be the neighbors of v with terminal verticesvg1, vg2, . . . , vg`. Each vertex vgj gives rise to the double coset HgjH.

Set B = ∪`i=1HgiH, D = (gH, gbH) | b ∈ B, g ∈ A and let the

dart-reversing involution be L(gH, gbH) = (gbH, Hg). Then the graphX is isomorphic to the coset graph C(A; H, B).

The above discussion generalizes the Sabidussi Theorem as follows.

Theorem 1.15. Every coset graph X = C(A; H, B) is vertex-transitive.Conversely, every connected simple vertex-transitive graph X is iso-morphic to a coset graph C(A; H,B), where A ≤ Aut(X) acts vertex-transitively on X, H is a stabilizer of a vertex, B = B−1 and A =〈H,B〉.

Note that for H = 1 the definitions of coset graph and Cayley graphcoincide.

Example 1.3. For example, let A = S5 and let

H = 1, (123), (132), (12), (13), (23), (45), (123)(45),

(132)(45), (12)(45), (13)(45), (23)(45)of order 12 which is isomorphic S1,2,3 × S4,5. Set a = (14)(25) andB = HaH. Then 〈H, a〉 generates S5, and C(A; H, B) is the Petersengraph GP (5, 2).

A graph X is called arc-transitive or dart-transitive if its automor-phism group acts transitively on darts. Clearly, all arc-transitive graphs


are vertex-transitive. But the converse is not true. A coset graphC(G; H, B) is arc-transitive if and only if B forms a single double coset.

The following ‘orbital graph’ construction gives another point ofview on vertex-transitive graphs.

Orbital graphs. If G ≤ Aut(X) is transitive on darts of a graphΓ = (D, V ; I, λ), then G acts on V transitively and induces a naturalaction on V ×V by (x, y)g = (xg, yg) for g ∈ G, in which D is an orbit.

Conversely, a transitive G-action on a set V induces an action onV × V by (x, y)g = (xg, yg) for g ∈ G. Its orbits are called orbitals.The diagonal ∆ is a trivial orbital; all others in ∆c = (V × V ) \ ∆are nontrivial. An orbital O is called self-paired if O = O−1, where(x, y) ∈ O−1 if and only if (y, x) ∈ O . For any self-paired orbital O,(O, V ; I, λ), where λ(x, y) = (y, x) and I(x, y) = x, is an arc-transitivegraph, called an orbital graph.

An orbital graph X = (O, V ; I, λ) determined by a transitive actionof G on V is isomorphic to a coset graph C(G; H, B), where A ≤Aut(X) acts vertex-transitively on X, H is a stabilizer of a vertex andB is generated by a single element a.

In this case we write O(G; H, a) for the orbital graph. This is infact isomorphic to C(A; H, HaH). Clearly, the graph is connected ifand only if G = 〈H, a〉. For instance O(S5; H, a) is the Petersen graphgiven in the previous example.

All connected arc-transitive graphs are orbital graphs up to isomor-phism.

Theorem 1.16. Every orbital graph X = O(G; H, a) is arc-transitive.Conversely, every connected simple arc-transitive graph X is isomor-phic to an orbital graph O(G; H, a), where A ≤ Aut(X) acts vertex-transitively on X, H is a stabilizer of a vertex and a ∈ A\H such thatA = 〈H, a〉.

Assume the orbital generated by a is non-self-paired. Then G actsvertex and edge transitively but not arc-transitively on the orbitalgraph X = O(G; H, a). Generally, an action of a group of automor-phisms of graph is called half-arc-transitive if it is edge and vertextransitive, but not arc-transitive. A half-arc transitive action of G ondarts of X has two orbits D = D+∪D−. We call D+ a G-orientation of


X. Since a G-orientation is balanced, each vertex has an even valency2k for some k ≥ 1.

The arc-transitivity of a graph X can be generalized obviously toa transitivity on the walks of a given length. We say that a groupG ≤ Aut(X) is t-transitive, t ≥ 1, if it acts transitively on the set ofirreducible walks of length t. The automorphism group of a cycle is t-transitive for any t. Surprisingly, with the exception of cycles, maximalt for which Aut(X) is t-transitive is bounded by 7 in general [We81],and by 5 in case of cubic graphs (see Tutte [Tu47]).

More precisely Weiss proved the following.

Theorem 1.17. (Weiss [We81]) Let X be a simple t-transitive con-nected graph of valency > 2. Then t ≤ 7. Moreover, a 6-transitivegraph is 7-transitive as well.

Weiss’s proof depends on the classification of finite simple non-abelian groups which is a major achievement of group theory of 20th

century. No independent proof is known yet.Highly symmetric graphs can be defined as incidence graphs of finite

geometries or block designs. Few examples follow.

Example 1.4. The Pappus graph 93 is 3-arc-transitive cubic graphon 18 vertices, being the incidence graph (D, V ; I, L) of the Pappusconfiguration

B = 123, 456, 789, 147, 258, 369, 158, 348, 267,which is a union of three parallel classes of lines in the affine geometryAG(2, 3), with exactly one set of three parallel lines missing. In otherwords, V = P ∪ B, where P = 1, 2, . . . , 9, (x, y) is a dart if x ∈ Pand x ∈ y ∈ B, or y ∈ P and y ∈ x ∈ B, I is the projection on the firstcoordinate and L(x, y) = (y, x). The automorphism group of 93 hasorder 216, and is a semi-direct product of a non-abelian group of order27 and exponent 3 by a dihedral group of order 8. Another remarkableproperty of 93 is that it has a highly symmetrical hexagonal embeddingin the torus, (the map 6, 33,0 in the notation of Coxeter and Moser,see Figure 1.4).


Figure 1.4: Hexagonal embedding of the graph of Pappus configurationin the torus

Example 1.5. The Heawood graph F014 is the incidence graph of theFano plane

P = 123, 345, 156, 147, 257, 367, 246,with automorphism group PSL(3, 2) : Z2

∼= PGL(2, 7). The graph is4-arc regular, and also admits a one-arc regular action of a subgroup oforder 42. There is a well-known hexagonal embedding of the Heawoodgraph, (the map 6, 32,1 in the notation of Coxeter and Moser; seeFigure 1.5).

Example 1.6. Vertices of the Coxeter graph F028 may be taken asantiflags of the Fano plane P (that is, ordered pairs (p, `) consisting ofa line ` and a point p not incident to `), and two vertices γ = (p, `) andδ = (q, m) are adjacent if P = `∪m∪p, q. The automorphism group


Figure 1.5: Hexagonal embedding of the Heawood graph in the torus

of the Coxeter graph has 336 elements and is isomorphic to PGL(3, 2) :Z2∼= PGL(2, 7). This graph is 3-regular.

Example 1.7. Constellations. Motivated by some investigations inlow-dimensional topology and theory of Riemann surfaces, the followingcombinatorial objects called constellations are frequently investigated.A constellation is a sequence of permutations [g1, g2, . . . , gk], where gi ∈Sn, G = 〈g1, g2, . . . , gk〉 is transitive on the set of n points and theproduct is g1g2 . . . gk = id. It follows that G has a right action onthe set of points. Two constellations C = [g1, g2, . . . , gk] and C ′ =[g′1, g

′2, . . . , g

′k] based on sets E and E ′, respectively, are isomorphic


if there is a bijection h : E → E ′ such that g′i = h−1gih, for i =1, 2, . . . , k. The automorphism group of a constellation is formed bythe permutations centralizing all gi, i = 1, 2, . . . , k. It follows that theautomorphism group of C is the centralizer of G in the symmetric groupSn. If gi act from right (left) then automorphisms act on E from left(right). Generally, G is much larger than Aut(C) and , by Theorem 1.7G ∼= Aut(C) if and only if |G| = |Aut(C)|.

Exercises

1.4.1. Find the least Cayley graph such that the left and right actions of Gon the vertex set are non-isomorphic.

1.4.2. Find all Cayley graphs with at most 8 vertices and distinguish themup to equivalence of group actions and up to graph isomorphisms.

1.4.3. Prove that the Petersen graph GP (5, 2) and the dodecahedral graphGP (10, 2) are not Cayley graphs.

1.4.4. Define the 1-skeletons of the five Platonic solids as orbital graphs.1.4.5. Prove that each octahedral graph On can be represented as a Cayley

graph based on a non-abelian group.1.4.6. For a coset graph G = G(A; H, B), show that

(1) A acts arc-transitively if and only if B = HbH, ∀b ∈ B.(2) If B = H`H for some ` ∈ A then G is arc-transitive of valency

|H|/|H ∩ `H`−1|.1.4.7. Prove Theorem 1.15.1.4.8. Find all non-isomorphic constellations on 6 points.1.4.9. Prove that the action of the automorphism group on the set of points

of a constellation is semi-regular.

Chapter 2

Coverings and liftings ofautomorphisms

2.1 Graph coverings and two group ac-

tions on the fibre

The aim of this section is to introduce a graph covering and two naturalgroup actions on the vertex set of the covering graph: One is by thecovering transformation group (from the left) and the other is by thefundamental groupoid (from the right). A graph covering is roughlyspeaking, an epimorphism between graphs which is locally bijective.A natural way to construct graph covering is to consider a projectionof a graph to its quotient graph defined by a semiregular group ofautomorphisms.

Definition 2.1. Let X = (D,V ; I, λ) and X = (D, V , I, λ) be graphs.A graph epimorphism p : X → X is called a covering projection if, forevery vertex u ∈ X, p maps the neighborhood Du of u bijectively ontothe neighborhood Dpu of pu. The graph X is usually referred to as thebase graph or a quotient graph and X is called the covering graph. Byfibu = p−1u and fibx = p−1x we denote the fibre over u ∈ V and x ∈ D,respectively.

23

24 Chapter 2. Coverings and liftings of automorphisms

Example 2.1. (Two quotients of the n-cube Qn) The n-cube or the n-dimensional Hypercube Qn = (D,V ; I, λ) is defined as a Cayley graphCay(Zn

2 , B), where V = Zn2 , the n-dimensional vector space over the

field (Z2, +, ·) and B = εi | i = 1, . . . , n is the standard basis for V =Zn

2 . That is, εi ∈ V is a vector of length n whose entries are all 0 exceptfor the i-th entry which is 1. All translations by elements in V form agroup of automorphisms t(V ) of Qn = Cay(Zn

2 , B), which acts regularlyon the vertex-set and semiregularly on the dart-set of Qn. One caneasily show that the quotient projection p : Qn → Qn/t(V ) is actuallya covering projection. Figure 2.1 shows a covering p : Q3 → Q3/t(V ),where the quotient graph Q3/t(V ) has one vertex and 3 semiedges. Ingeneral, the quotient Qn/t(V ) has one vertex and n semiedges, and itcan be described as Qn/t(V ) = (D, V ; I , λ), where V = 0, D = Zn,I : Zn → 0 and λ = id. This graph is called the n-semistar anddenoted by stn.

Another instance of this situation one can get a quotient graph bythe antipodal mapping x 7→ −x on V . The quotient graph is called thehalved cube. In particular if n = 3 the antipodal action on Q3 definesthe quotient graph isomorphic to the complete graph K4 on 4 vertices.

000

111222

333

444

555 666

777

a0b0 c0

a1

c1b1 a2

b2

c2

a3b3

c3

a4b4

c4

a5

b5

c5

a6

b6c6

a7

b7

c7

a

b

c

Figure 2.1: A covering projection Q3 → st3, given by xi → x

2.1. Graph coverings and Two actions 25

Example 2.2. (The n-cube covers the n-dipole) For another cover-ing projection from the n-cube, we use the same notation as in Ex-ample 2.1. Let T : Zn

2 → Z2 be a linear transformation defined by(x0, x1, x2, . . . , xn−1) 7→

∑n−1i=0 xi. Then the kernel K := Ker(T ) con-

sists of the vectors expressible as a sum of even number of vectors in thestandard basis B for V = Zn

2 . All translations by elements in K forman automorphism group t(K) of Qn, which acts semiregularly on thevertex and dart sets of cube Qn. Note that the translation group t(K) isisomorphic to Zn−1

2 , generated by t(ε0+ε1), t(ε1+ε2), . . . , t(εn−2+εn−1),where εi ∈ B. We may form a quotient graph described by Qn/t(K) =(D, V ; I , λ), where V = Z2, D = Z2 × Zn, the incidence function Iis the projection on the first coordinate and the dart-reversing involu-tion λ takes (i, j) 7→ (i + 1, j). This graph is called the n-dipole anddenoted by D(n). It has two vertices and joined by n parallel edgesDefine p : Qn → D(n) by p(a, εi) = (0, i) if a ∈ K, and p(a, εi) = (1, i),otherwise. Clearly p : Qn → Qn/t(K) is a covering projection. Wehave fib0 = K, fib1 = Zn

2 \ K, fib(0,εj) = (a, εj) | a ∈ K andfib(1,εj) = (a, εj) | a ∈ Zn

2 \ K for each j ∈ Zn. See Figure 2.3.

Consider an arbitrary dart x ∈ X with its initial vertex u. Bydefinition, for each vertex u ∈ fibu there exists a unique dart x initiatingat u which is the lift of x, that is, px = x. This means that everywalk of length 1 starting at u lifts uniquely to a walk starting at u.Consequently, by induction we obtain the following “Unique walk liftingtheorem”. (For the classical topological variant, see [Ma91, pp. 151].)

Proposition 2.1. [Walk lifting theorem] Let p : X → X be acovering projection of graphs and let W : u → v be an arbitrary walkin X. Then:

(1) For every vertex u ∈ fibu there is a unique walk W which projectsto W and has u as the initial vertex.

(2) Homotopic walks lift to homotopic walks.

(3) If X is connected, then the cardinality of fibu does not depend onu ∈ X.


Proof. (1) Use induction on the length of W . If |W | = 1 then thestatement holds by the definition of p. Let W = W1x where x is adart. By the induction hypothesis W1 lifts to a unique walk W1 withan initial vertex u and a terminal vertex v. Then W lifts to the uniquewalk W1x, where x is the unique lift of x based at v.

(2) Let P and Q be homotopic walks. It is enough to prove thestatement for the case when Q arises from P by cancelling or addinga subsequence of the form xx−1. Let Ix = v and let v be the lift of vtraversed by the unique lifts P and Q. By part (1) xx−1 lifts to a uniquev-based closed walk xx−1. It follows that P and Q are homotopic.

(3) Since X is connected, for any two vertices u, v there exists a walkW joining u to v. Assume W = x is of length 1. Then there are |fibu|lifts of x. The terminal points of these lifts are all distinct, otherwisewe get a pair of lifts of x−1 based at the same vertex contradicting part(1) . Hence |fibu| ≤ |fibv|. Replacing W by W−1 we get |fibv| ≤ |fibu|,hence |fibu| = |fibv|. Since X is connected, by induction the statementextends to a pair of vertices at any distance. 2

Note that Proposition 2.1 may fail when considering a path startingat the free endpoint of a semiedge in the 1-CW complex associated witha graph. This difficulty does not occur with combinatorial graphs sincewe only consider walks which start and end at vertices. It followsfrom Proposition 2.1 that the cardinality |fibu| of a covering projectionp : X → X, where X is connected, does not depend on the vertex u. If|fibu| = n, we say that the covering projection p is n-fold.

Let p : X → X be a covering projection. To simplify the notation wewrite π = π(X) and πu = π(X, u). As an immediate consequence of theunique walk lifting we have the following simple but useful observation.

Proposition 2.2. Let p : X → X be a covering projection of graphs.Then there exists a right action of the groupoid π on the vertex set ofX defined by

u ·W = v,

where W : pu → pv is a walk in π and v is the endvertex of the unique(lifted) walk W over W starting at u. Moreover, u 7→ u·W is a bijectionfibpu → fibpv.


This right action of the fundamental groupoid π (or the fundamentalgroup) on the vertex set of X is known as the monodromy action.

In view of Proposition 2.1, the action of the fundamental groupoidcan, in fact, be extended to an action of all walks. In particular, thefundamental group πu acts on fibu. If X (and hence X as well) isconnected, then this action is transitive. To simplify the notation wedenote by πu = (πu)u the stabilizer of u ∈ fibu under the action ofπu. In this case, the stabilizer πu is the image of πu by the grouphomomorphism p∗ : πu → πu.

In fact, one can prove the following.

Proposition 2.3. Let p : X → X be a covering of connected graphs.Then the mapping p∗ : π(X, u) → π(X, u) defined by W 7→ pW = W(W a closed u-based walk) is a monomorphism of groups.

Conversely, let X be a connected graph and G ≤ π(X, u) be a sub-group of finite index. Then there is a connected graph X and a coveringp : X → X such that G = p∗(π(u, X)).

As an application we can prove the following theorem by Schreier.

Theorem 2.4. Let Fβ be a free group of rank β. Let H < Fβ be asubgroup of index m. Then H is a free group of rank m(β − 1) + 1

Proof. By Lemma 1.2 Fβ can be identified with a fundamental groupof a one-vertex graph X with β loops. By Proposition 2.3 H is afundamental group of a graph p : X → X covering X. Hence H = π(X)is a free group of rank V (X)−E(X)+1 = n−βn+1 = n(β−1)+1, wheren is the number of folds. Since p∗ : π(X) → π(X) is a monomorphism,n = m and we are done. 2

Clearly, the actions of the fundamental groups at two distinct ver-tices of a connected graph are isomorphic. To show this, let W ∈ π bea walk from u to v. By Proposition 2.2 W : u 7→ u ·W is a bijectionand W∗ : πu → πv defined by S 7→ W−1SW is an isomorphism. Then(W , W∗) : (fibu, π

u) → (fibv, πv) is an isomorphism of the actions.

Another important group action on the fibre of a covering projec-tion p : X → X is defined with the ‘covering transformation group’.


An automorphism ϕ of X satisfying pϕ = p is called a covering trans-formation. The set of all covering transformations forms a group underthe composition, denoted by CT(p) and called the covering transfor-

mation group. This group acts on X from the left. In fact, it acts oneach fibre fibu. By definition, the (right) π-action on the vertex set ofX defined in Proposition 2.2 commutes with the (left) action of CT(p).Each fibre is fixed by CT(p) set-wise. Thus a group action of the

covering transformation group on is defined on each fiber.

As an immediate consequence of the unique walk lifting, we havethe following observation.

Proposition 2.5. Let p : X → X be a covering projection of con-nected graphs. Then CT(p) acts semiregularly on X; that is, CT(p)acts without fixed points both on vertices and on darts of X.

Proof. It is enough to show that f ∈ CT(p) has no fixed vertices exceptthe identity. Suppose that f(x) = x. We need to show f(y) = y for anyvertex y in X. To do this, take a walk W from x to y. The uniquenessof walk lifting implies f(W ) = W , and hence f(y) = y. 2

By Proposition 2.5, every covering transformation f ∈ CT(p) isuniquely determined by the image of a single vertex of X. The factthat CT(p) acts semiregularly for a connected covering p is particularlyimportant. Therefore, for the rest of this section, as well as for most ofthe book, the graphs X and X will be assumed to be connected.

By the semiregularity of CT(p) we have |CT(p)| ≤ |fibu|, with equal-ity occuring if and only if CT(p) acts regularly on each fibre. The cov-ering projection is called regular if CT(p) acts regularly on each fibre.

In Figure 2.2, the projection (b) is regular, but (a) is not.

There is an alternative approach to regular coverings of graphs.Given a connected graph X = (D,V ; I, L) let G ≤ Aut(X) be a sub-group acting semiregularly on the both sets D, V of darts and verticesof X. The regular quotient X/G = (D, V ; I , L) is defined by settingD = [x]G|x ∈ D, V = [v]G|v ∈ V , I[x]G = [Ix] and L[x]G = [Lx]G.


w3

w2

w1

u1

u2

u3

v1

v2

v3

w2

w1

w0

u

u0 v0

v1

v2

? ?

u1

v

(a)

w

(b)

w

u v

u2

Figure 2.2: Regular and irregular graph coverings

Proposition 2.6. (1) Let X = (D, V ; I , λ) be a connected graph andlet A ≤ Aut(X) act semiregularly on the vertex set V and also onthe dart set D. Then the mapping q : X → X/A taking x 7→ [x]Ais a regular covering projection with CT(q) = A.

(2) Any regular covering projection p : X → X between connectedgraphs is equivalent to a covering q : X → X/CT (p) to the regularquotient, that is, there is a graph isomorphism α : X/CT (p) → Xsuch that α q = p.

Proof. (1) The semiregularity of A-action on the vertices and darts ofX implies that q is a covering X → X. The elements of A permute thedarts in fib[x] = [x] for each x ∈ D. By definition, A acts regularly on[x]A, implying CT(q) = A.

(2) If p : X → X is regular then A = CT (p) acts regularly on thefibres over each vertex and each dart. It follows that the fibres areformed by the orbits of A acting semiregularly on vertices and darts.Hence there is a one-to-one correspondence between the orbits of Aon V and D and elements of V and D, respectively, determining therequired graph isomorphism α. 2


Hence, the coverings in Examples 2.1, 2.2 are all regular with thecovering transformation groups Zn

2 and K C Zn2 , respectively.

Exercises

2.1.1. Prove that |Aut(Qn)| = 2nn! and it is transitive on vertices, arcs and2-arcs but not on 3-arcs. Hint: Aut(Qn) ∼= Zn

2 o Sn consisting of the(left) translations of V = Zn

2 and the permutations of n coordinates.2.1.2. Let p : X → X be a covering of graphs. Prove that the mapping

p∗ : π(X, u) → π(X,u) defined by W 7→ pW = W (W a closed u-based walk) is a monomorphism of groups.

2.1.3. * Let X be a connected graph and G ≤ π(X, u) be a subgroup.Prove that there is a graph X and a covering p : X → X such thatG = p∗(π(u, X)).

2.1.4. * An inclusion of the trivial group into the fundamental group of aconnected graph X induces a so-called universal cover over X. Theuniversal cover X over X is characterized by the following property:if Y → X is a covering of connected graphs then X covers Y .

Describe a procedure constructing the respective covering graph. De-scribe the universal cover over a two vertex graph, where the twovertices are joined by three internally disjoint paths of lengths 1, 2and 3.

2.1.5. For which graphs the universal cover is a finite graph? (countablegraph ?)

2.1.6. Prove Theorem 2.14.2.1.7. Given covering p : X → X the (right) monodromy action of the funda-

mental group on a fibre fibu determines a permutation representationof π(u,X)/Core(p∗π(v, X)) called the monodromy group.

Prove that in a regular covering the monodromy group and group ofcovering transformations are isomorphic.

2.2 Voltage spaces

The goal of this section is to explain, unify and generalize the classicalconcepts of ordinary, permutation and relative voltage assignments in-troduced by Alpert, Biggs, Gross, Tucker and others [GT87] in the1970’s. In fact, this unified approach works in the general setting

2.2. Voltage spaces 31

of fairly arbitrary topological spaces, see [Ma98]. Compare also with[Su84].

A voltage space on a connected graph X is a triple (F, G; ξ) where Gis a group acting on a nonempty set F (from the right) and ξ : π → Gis a homomorphism from the groupoid π = π(X) to G. The group Gis called the voltage group, F is the abstract fibre and ξW is the voltageof a reduced walk W . Since the product of reduced walks is mappedby the homomorphism ξ to the product of voltages, it follows that anytrivial walk carries the trivial voltage and, consequently, inverse walkscarry inverse voltages.

In order to define a voltage space on a graph it is obviously enoughto specify the voltages on walks of length 1. Even more, we only needto specify the voltages on darts ξ : D → G subject to the condition

ξx−1 = (ξx)−1.

Therefore, ξ is determined by the values on an orientation D+ ⊂ D. Byinduction one can then extend the assignment of voltages to all walks,not just to the reduced ones. Clearly, homotopic walks carry the samevoltage. Note that a voltage associated with a semiedge is always aninvolution.

If the voltage group acts faithfully, we speak of a monodromy voltagespace; by taking the permutation representation of a faithful action weget the permutation voltage space as its canonical representative. If thevoltage group acts regularly, we speak of a regular voltage space. Inparticular, if G acts on F = G by right translation, we speak of theCayley voltage space (ordinary voltages in the terminology of Gross andTucker [GT87]).

Obviously,Gu := ξ(πu)

is a group, called the local voltage group at the vertex u. Moreover, letW : u → v be a walk. Then the inner automorphism

W#(g) = ξ−1W gξW

of G takes Gu onto Gv. A voltage space is called locally transitivewhenever a (and hence all) local voltage group acts transitively on theabstract fibre.


We say that a voltage space (F,G; ξ) on a graph X is associated witha covering projection p : X → X if there exists a labelling ` : V → F ofthe vertex set V of X such that, for each vertex u of X, the restrictions

ù = `|fibu : fibu → F

are bijections and moreover,

(`, ξ) : (V , π) → (F,G)

is a morphism of actions. In other words, for every (reduced) walkW : u → v of X and for each u ∈ fibu we have

`(u ·W ) = ù · ξW .

(For simplicity, we use the same dot sign for the action of π and theaction of G.) Two coverings p1 : X1 → X and p2 : X2 → X are saidto be isomorphic if there exists a graph isomorphism Φ : X1 → X2

such that p1 = p2Φ. Voltage spaces associated with isomorphic cov-ering projections are called equivalent. Note that the action of ξW

on F “represents” the bijection W : u 7→ u · W from fibu to fibv

introduced as the monodromy action in Proposition 2.2: indeed, itspermutation representation W ∈ SymL F is precisely the permutationξW ∈ SymR F . Furthermore, from the connectedness, one can showthat (ù, ξ) : (fibu, π

u) → (F, Gu) is an epimorphism of actions with ù

being a bijection. Hence Gu acts on F “in the same way modulo rela-belling” as πu does on fibu, and thus both actions have “the same localmonodromy”. In fact, it is the local group which is responsible for thestructure of the covering. In particular, a voltage space associated witha covering projection defined on a connected graph is necessarily locallytransitive, and as in the classical case [Sk86], a covering projection isregular if and only if Gu has a normal stabilizer.

Each covering projection of graphs gives rise to an associated voltagespace, for instance to the permutation voltage space (cf. [GT77]).

Theorem 2.7. Let p : X → X be a covering projection between con-nected graphs. Then there exists a locally transitive permutation voltagespace associated with p.

Conversely, every locally transitive voltage space (F, G; ξ) on a graphX derives a covering projection X → X whose associated voltage space


is (F,G; ξ). Such a covering is determined up to isomorphism, and ifX is connected then so is X.

We denote by Xξ → X a covering projection derived from a voltagespace (F,G; ξ) on a graph X.

Proof. We set F = fibb, where b is a fixed vertex. By Proposition 2.1(3)each fibre fibv has the same cardinality. Take any labeling `v : F → fibv.For each walk W : u → v, we set the voltage ξW to be the permutationW ∈ SymR F representing the bijection W . That means `(u)·ξW = `(v)if and only if u · W = v. Now the voltage group G ≤ SymR F isgenerated by the voltages ξW , where W ranges through all walks inX. Since X is connected, for any two vertices u1, u2 ∈ fibb there is a(reduced) walk W : u1 → u2. It follows that there is a u-based closedwalk W = p(W ) such that u1 ·W = u2. Hence, `(u1) · ξW = `(u2) andGu is transitive on fibu. The labelling ` and the voltage assignment ξis defined such that for each u ∈ fibu we have `(u ·W ) = `u · ξW . Hence(`, ξ) : (V , π) → (F,G) is a morphism of actions.

Conversely, given locally transitive permutation voltage assignment(F, G, ξ) on X = (D, V ; I, λ) we define the derived covering graph X =(D, V , I, λ) as follows. For u ∈ V or x ∈ D we set fibu = u × Fand fibx = x × F . Hence V = V × F and D = D × F . We setI(xi) = (Ix)i and λ(xi) = λ(x)j, where j = i · ξx. The mappingp : xi 7→ x takes surjectively D onto D and it is locally bijective. LetI(x) = u. We have p(I(xi)) = p(ui) = u = I(x) = I(p(xi)). Further,p(λ(xi)) = p(λ(x)j) = λ(x) = λ(p(xi)). It follows that p is a coveringX → X. Moreover, the natural projection ` : ui 7→ i determines amorphism of actions. Indeed, the equality `(u ·W ) = `u · ξW holds bydefinition of λ for walks of length one, and it extends (by induction) towalks of any length.

If X is connected and (F, G, ξ) is locally transitive then X is con-nected. Indeed, any two fibres fibu, fibv are joined by a lift of a walk Wjoining u to v in X. Since (F,G, ξ) is locally transitive for any i, j ∈ F ,there is an element ξW in the local group Gu with W ∈ πu, such thati · ξW = j. The unique lift of W with initial vertex ui joins ui to uj.Consequently, X is connected.

The isomorphism problem is solved in Theorem 2.12. 2


The lift X, as well as the covering p : X → X determined by avoltage space (F, G, ξ) will be called the derived graph, and the derivedcovering, respectively. In particular, if the covering projection p is n-fold we may assume that fibu = u1, . . . , un for each vertex u of X andthat the label of the vertex ui is i ∈ 1, . . . , n. Then the permutationvoltage ξx ∈ Sn of a dart x joining the initial vertex u to the terminalvertex v = Ix−1 is defined by the following rule: ξx(i) = j if and onlyif there is a dart x ∈ fibx joining ui to vj.

With a regular covering projection we can associate a Cayley voltagespace (G,G; ξ) as follows. First label the elements of every vertex fibreby elements of G ∼= CT(p) so that the left action of CT(p) is viewed asthe left translation on itself. Then the voltage action of G is the righttranslation on itself, that is, W (g) = gξW (cf. [Gr74]). We have thefollowing theorem (compare with [GT87, pp. 57–71]).

Theorem 2.8. Let p be a regular covering projection between connectedgraphs. Then there exists an associated locally transitive Cayley voltagespace.

Conversely, every locally transitive regular (in particular, Cayley)voltage space (F,G; ξ) on a graph X derives a regular covering projec-tion X → X whose associated voltage space is (F, G; ξ). Such a coveringis determined up to isomorphism, and if X is connected then so is X.

We denote by X ×ξ G → X a regular covering projection derivedfrom a Cayley voltage space (G,G; ξ) on a graph X.

Proof. We present only the constructive part of the proof. First recalla regular action of G on F is equivalent to a Cayley action of G on itselfby right translation. Hence we may assume that the voltage space is aCayley voltage space (G,G; ξ) is defined on a graph X = (D, V ; I, λ).Define X = (D, V ; I , λ) by setting D = D ×G = xk | x ∈ D, k ∈ G,V = V × G = vk | v ∈ V, k ∈ G, I(xk) = I(x)k and λ(xk) =(λx)kξ(x). Then the first coordinate projection X → X is a covering

projection associated with (G,G; ξ) and the graph X is the derivedgraph determined by the Cayley voltage space (G,G; ξ).

The covering isomorphism problem is settled in the next section,see Theorem 2.14. 2


↓

0 1

000

101

110

011

100

001

010

111

010

-

--100001

Figure 2.3: Q3 covers D(3)

Example 2.3. (Cayley graphs are regular covers of monopoles). Let uscall a graph with one vertex and arbitrary number of darts a monopole.Recall the definition of the Cayley graph. Given a group G and a sym-metric generating (multi)set S = S−1 of G the Cayley graph Cay(G,S)is the graph (G×S,G; IS, λS), where IS = pr1 is the projection onto thefirst factor and the involution on darts is given by λS(g, s) = (gs, s−1).(Note that S being possibly a multiset allows Cayley graphs to haverepeated generators.) Consider the monopole X = (S, v; I, λ) whereI is the constant function and λ(s) = s−1. It is easy to see that theprojection onto the second factor pr2 : G×S → S extends to a regularcovering projection Cay(G,S) → X.

Conversely, let (G,G; ξ) be an arbitrary Cayley voltage space ona monopole X = (D, v; I, λ). Then the derived covering graph is theCayley graph Cay(G,S) where S = ξx | x ∈ D is the corresponding


symmetric (multi)set of generators for the group G.This shows that Cayley graphs are nothing but regular covers of

monopoles. Summing up, by allowing semiedges we have obtained acharacterization of Cayley graphs which overcomes the trouble encoun-tered in the classical approach that not all Cayley graphs were regularcoverings of bouquets of circles [GT87, pp. 68–69, 272].

Observe that the voltage group can be larger than really necessary.The following proposition shows that any given voltage space can bereplaced by an equivalent voltage space where the local groups do notdepend on the vertex and coincide with the whole voltage group.

Proposition 2.9. Let (F,G; ξ) be a voltage space associated with acovering projection p : X → X of connected graphs, and let b ∈ X be abase vertex. Then there exists an equivalent voltage space (F, G; ξ′) allof whose local voltage groups are the same and the local group at b hasnot changed.

Proof. For each vertex u ∈ X choose a preferred walk W u : u → b,where W b is the trivial walk at b. This can easily be done, for instance,by specifying a spanning tree T for X and choose W u : u → b asa walk in T . Let W : u → v be an arbitrary walk in X. Defineξ′W = (ξW u)−1ξW ξW v . Then, the new voltage ξ′ of each preferred walkis trivial and ξ′S = ξS for every closed walk S ∈ πb. Now, modify thelabelling of fibres according to the rule `′t = `t · ξW u , t ∈ X. It is easyto see that the new voltage space has all the claimed properties. 2

Voltage space constructed in the proof of Proposition 2.9 will becalled T -reduced. Its characteristic feature is that all the tree-darts areassigned by a trivial voltage.

Since any two vertices can be joined by a walk carrying the trivialvoltage, no walk can have voltage outside Gb. For if a walk W : u → vhad a voltage not in Gb, then the closed walk S = (W u)−1WW v at bwould have its voltage ξ′S = ξ′W 6∈ Gb, which is absurd. Hence, the localgroup Gb can be chosen as the new voltage group without affecting thecovering. Therefore, assuming that a voltage space (F, G; ξ) satisfies

Covering isomorphism problem 37

the condition Gb = G does not loss of any generality, but can some-times considerably simplify the notation. Hence we have the followingcorollary.

Corollary 2.10. Let p : X → X be a (regular) covering of connectedgraphs. Then there is an associated permutation (Cayley) voltage space(F, G; ξ) such that G = Gu for any vertex u of X.

Exercises

2.2.1. Prove that generalized Petersen graphs GP (n, k) can be defined bymeans of a Cayley voltage space on a 2-vertex graph.

2.2.2. Find all regular quotients of the 3-dimensional cube and determinethe respective Cayley voltage spaces.

2.2.3. Construct all 4-fold coverings over a bouquet of two cycles and dis-tinguish them up to isomorphism.

2.2.4. Prove that a Cayley voltage space (G,G; ξ) on a connected graph Xdetermines a derived covering X → X with X connected if and onlyif every local group Gb = G.

2.2.5. A Cayley voltage space (Z2,Z2; ξ) with ξ(x) = 1 ∈ (Z2, +) determinesa canonical double cover X over X. Show that X is connected if andonly if X is not bipartite.

2.2.6. Let X be a connected graph with v vertices and e edges. Let k =e−v+1, G = Zk

2 and T be a spanning tree of X. Let x1, x2, . . . , xk bea set of cotree darts such that xj 6= x−1

i for any i 6= j. Let ξ(xi) ∈ Gforms a generating set for G.

(1) Prove that the 2k cover over X defined by the Cayley voltagespace (G,G, ξ) is connected.

(2) Describe the kernel of the associated epimorphism W → ξW fromthe fundamental group π(X) → G.

2.2.7. * Prove Theorem 2.8.

2.3 Covering isomorphism problem

We define a lifting of any graph morphism as follows: For a givenmorphism f : Y → X of graphs and a covering projection p : X → X,if there exists a morphism f : Y → X such that the diagram


X

p

²²Y

f??ÄÄÄÄÄÄÄ f // X

commutes, then f is called a lifting of f . One may ask naturally whenthe morphism f can be lifted?

The lifting problem shall be concerned with the following two direc-tions in this book. Firstly, it is related to the isomorphism problem ofthe covering projections of a given graph. Secondly, given covering pro-jection p between two graphs we will investigate which automorphismsof the base graph lift along p. In this section we concentrate to thecovering isomorphism problem while the lifting automorphism problemwill be discussed in the following sections.

Theorem 2.11. [Lifting Criterion] Let p : X → X be a coveringprojection between connected graphs and let f : Y → X be a morphismof graphs. Let Y be connected. Then, the morphism f has a lift f :Y → X if and only if for any vertex y0 ∈ Y , there exists a vertexx0 ∈ fibf(y0) such that f∗(π1(Y, y0)) ⊂ p∗(π1(X, x0)).

µ

?

-

f

x0

ρ g(y)

X

f σσ

σ2

y

y0

Y Xf

p

f(y)

x0f σ2

Figure 2.4: Lifting of f

2.3. Lifting and covering isomorphism problems 39

Proof. (⇒) is easy: If f lifts to f , then

f∗(π1(Y, y0)) = (p f)∗(π1(Y, y0)) = (p∗ f∗)(π1(Y, y0)) ⊂ p∗(π1(X, x0)).

(⇐): Suppose that

f∗(π1(Y, y0)) ⊂ p∗(π1(X, x0)).

Then a lift f : (Y, y0) → (X, x0) of f can be defined as follows: Forany vertex y ∈ Y , choose a walk σ from y0 to y. Then f σ is a walkfrom the vertex x0 to a vertex x = f(y). Now, by the “Unique walklifting theorem” (see Proposition 2.1), the walk f σ lifts uniquely toa walk starting from the vertex x0. Say ρ for such lifting walk. Now,define f(y) as the end vertex of ρ. To show that f(y) is well defined,we need to prove that show f(y) is independent of a choice of a walk σfrom from y0 to y. To do this, let σ2 be another walk from from y0 toy. Then, σ · σ−1

2 is a closed walk at y0 and [σ · σ−12 ] ∈ π1(Y, y0). By the

hypothesis, a closed walk (f∗[σ]) · (f∗([σ2])−1 in (X, x0) can be lifted to

a closed walk in X based at x0. Hence, the liftings of f σ and f σ2

have the same end vertex. Hence, the lifting f(y) is well defined. 2

Let Γ be a group of automorphisms of the graph X. Two coveringspi : Xi → X, i = 1, 2, are said to be isomorphic with respect to Γ if thereexist a graph isomorphism Φ : X1 → X2 and a graph automorphismγ ∈ Γ such that the diagram

X1 X2

X X

-

-

Φ

γ?

p1

?

p2

commutes. Or, equivalently, if there exists a γ ∈ Γ such that γp1 canbe lifted to an isomorphism Φ. Such a lifting Φ is called a coveringisomorphism with respect to Γ. Note that for any group Γ of automor-phisms of G, the covering isomorphic relation with respect to Γ on theset of coverings of G is an equivalence relation. As a labelled version ofthe base graph, the group Γ is often taken as the trivial group.


When Γ is trivial, we simply say that two covering projections areisomorphic. In particular, when X1 = X2 with p1 = p2 Φ is a coveringtransformation.

The number β(X) = |E(X)|−|V (X)|+1 is called the Betti numberof X.

Let T be a spanning tree of a connected graph X. Recall that apermutation voltage assignment φ is said to be T -reduced if φ assignsthe identity to the arcs of T . Let CT (X; r) denote the set of T -reducedpermutation voltage spaces (1, 2, . . . , r, Sym(r), φ), on X. In orderto find an algebraic property when two r-fold coverings p : Xφ → Xand q : Xψ → X are isomorphic, we first observe that an isomorphismΦ : Xφ → Xψ of the coverings takes the fibre over a vertex v ontothe fibre over v. Thus, Φ|p−1(v) : p−1(v) → q−1(v) is a permutationon the r vertices v1, v2, . . . , vr for each v ∈ V (X) and it can beconsidered as a permutation on the index set F = 1, 2, . . . , r. Now,we define f : V (X) → Sym(r) by f(v) = Φ|p−1(v) for v ∈ V (X)so that Φ(u, h) = (u, f(u)(h)) for each vertex (u, h) of the coveringgraph Xφ. Now, for a dart uv of X, if (u, h) is adjacent to (v, k)in Xφ, then φ(uv)(h) = k and Φ(u, h) = (u, f(u)(h)) is adjacent toΦ(v, k) = (v, f(v)(k)) in Xψ. Thus, we have ψ(uv)f(u) = f(v)φ(uv),or ψ(uv) = f(v)φ(uv)f(u)−1 for all uv ∈ D(X). The converse is left tothe reader as an exercise.

Theorem 2.12. Let X be a graph and φ, ψ be permutation voltageassignments defined on X.

Two r-fold coverings p : Xφ → X and q : Xψ → X are isomorphicif and only if there exists a function f : V (X) → Sym(r) such thatψ(uv) = f(v)φ(uv)f(u)−1 for each arc uv.

Moreover, if φ and ψ are T -reduced (with respect to some spanningtree T ) then the coverings p and q are isomorphic if and only if thereexists a permutation σ ∈ Sym(r) such that ψ(uv) = σφ(uv)σ−1 foreach arc uv not in T .

Given graph X choose a spanning tree T and an orientation D+.Set β = β(X). Denote Iso (X; r) the number of r-fold coverings of agraph X and by Isoc (X; r) denote the number of connected ones. Letβ be β(X). Then a T -reduced permutation voltage assignment canbe considered as a β-tuple of permutations in Sym(r) by labeling the

2.3. Lifting and covering isomorphism problems 41

positive darts in D(X) − D(T ) as e1, e2, . . . , eβ, and the set CT (X; r)can be identified as

CT (X; r) = Sym(r)× Sym(r)× · · · × Sym(r), (β factors).

We define an Sym(r)-action on the set CT (X; r) by simultaneous co-ordinatewise conjugacy: For any g ∈ Sym(r) and any (σ1, . . . , σβ) ∈CT (X; r),

g(σ1, σ2, . . . , σβ) = (gσ1g−1, gσ2g

−1, . . . , gσβg−1).

It follows from Theorem 2.12 that two T -reduced permutation voltageassignments φ and ψ in CT (X; r) yield isomorphic coverings of X if andonly if they belong to the same orbit under the Sym(r)-action. Hence,one can have the following theorem from the orbit-counting theoremcommonly known as Burnside’s Lemma.

Theorem 2.13. The number of r-fold coverings of X is

Iso (X; r) =∑

`1+2`2+···+r`r=r

(`1! 2

`2`2! · · · r`r`r!)β−1

.

An algebraic characterization of two isomorphic graph coveringsgiven in Theorem 2.12 can be rephrased for regular coverings as fol-lows [HKL96, Sk86].

Theorem 2.14. [HKL96, Sk86] Two connected regular coverings Xξ1 →X and Xξ2 → X derived from T -reduced Cayley voltage assignments(A,A, ξi), i = 1, 2 on X are isomorphic if and only if there exists anautomorphism σ ∈ Aut(A) such that σ ξ1 = ξ2.

Exercises

2.3.1. Compute the number of 4-fold coverings over a 3-dipole and the num-ber of regular 4-fold coverings and construct all the regular ones.

2.3.2. Under what condition two semi-regular subgroups of the automor-phism group determine isomorphic projections onto the respectivequotients?


2.4 Lifting automorphism problem - clas-

sical approach

Structural properties of various mathematical objects are to a largeextent reflected by their automorphisms. It is therefore natural tocompare related objects through their automorphism groups. Unfor-tunately, the automorphism group of an object is not a functorial in-variant in general. This excludes relatively easy comparison of therespective automorphism groups via morphisms. Much better chancesto get relevant results occur when the general problem is relaxed. Usu-ally, one considers a morphism m : Y → X satisfying some additionalproperties and asks whether associated with an automorphism f of Xthere exists an automorphism f of Y such that the diagram

Y Y

X X

-

-

f

f?m

?m

commutes. This is the essence of the problem of lifting automorphisms.Its most typical, well studied, but somewhat more general instanceis lifting continuous mappings along topological covering projections.Nevertheless, specific questions about lifting automorphisms receivedless attention.

The automorphism lifting problem has recently appeared in the con-text of graph coverings and maps on surfaces. Due to a discrete natureof these objects, their covering projections are usually treated combi-natorially rather than topologically. We are thus led to examine thelifting problem from this point of view as well.

We have seen that for a covering projection p : X → X, any walk inX can be lifted to a walk in the covering graph X. Also every coveringtransformation is a lifted automorphism ϕ of the identity automorphismof X which means p ϕ = idX p.

Let p : X → X be a covering projection of graphs and let f be anautomorphism of X. We say that f lifts if there exists a morphism fof X, called a lift of f , such that the diagram

2.4. Lifting automorphism problem - classical approach 43

X X

X X

-

-

f

f?p

?p

commutates. When f is a lift of f , we say f projects onto f . Later it willbe shown in that if f lifts and f has a finite order in Aut(X), so does f−1

and hence a lift of an automorphism should be an automorphism. Moregenerally, let A ≤ Aut(X). Then the lifts of all automorphisms in Aform a group, the lift of A, often denoted by A ≤ Aut(X). In particular,the lift of the trivial group is the covering transformation group CT(p).There is an associated group epimorphism pA : A → A with CT(p) asits kernel. The set of all lifts Lft(f) of a given f ∈ Aut(X) is a cosetof CT(p) in A.

By Proposition 2.5, every covering transformation f ∈ Lft(id) isuniquely determined by the image of a single vertex of X. In fact, thisis true for all lifts of an automorphism of the base graph X.

Proposition 2.15. Let p : X → X be a covering projection of con-nected graphs and let f ∈ Aut(X) have a lift. Then each f ∈ Lft(f) isuniquely determined by the image of a single vertex (as well as by theimage of a single dart) of X.

Proof. Let f and g be two lifts taking a vertex x to the same vertexy. Then f g−1 belongs to CT(p) and fixes x. It follows that f g−1 = 1implying f = g. 2

Every lifted automorphism in a covering projection p : X → Xpreserves the fibres over darts and also the fibres over vertices setwise.Thus the fibres form a system of blocks of imprimitivity of the actionof the group of all lifts.

Example 2.4. (A regular quotient Kn → Kn/Zn) Let X = Cay(Zn;Zn−0) ∼= Kn and let X = (D, V ; I, λ) be a monopole defined by D =Zn − 0, V = u, I : i 7→ u for i = 1, . . . , n − 1 and λ(i) = −i.Setting p(j, i) = i a covering p : X → X from Kn onto the monopole isdefined. We have CT(p) ∼= Zn. Set m = bn−1

2c. The monopole graph


X consists of bn−12c = m loops, and one semi-edge if n is even. The

pairs of darts forming loops form a system of blocks, and for each loopthere is an automorphism interchanging the two darts. It follows that|Aut(X)| = 2m ·m!. However |Lft(p)| ≤ |CT(p)|·|Aut(X)| = n2m ·m! <n! for every n ≥ 4. Employing the following Lemma 2.16 one can provethat Lft(p) is the Frobenius group which is isomorphic to Zn o Z∗n.

Hence the size of the automorphism of a covering graph may bemuch larger then the group of of lifts.

Example 2.5. (Canonical double coverings:) Let X = (D, V ; I, λ)be a graph. Its canonical double covering is defined by setting D =D × Z2, V = V × Z2, I(xi) = (I(x))i and λ(xi) = λ(x)i+1. It canbe easily seen that the mapping xi 7→ x determines a regular coveringp : X → X. The nontrivial element of CT(p) takes xi 7→ xi+1, and everyautomorphism ψ ∈ Aut(X) lifts to two automorphisms ψj, j ∈ Z2,defined by ψj(xi) = ψ(x)j+i, for i ∈ Z2. Hence, |Aut(X)| ≥ 2|Aut(X)|.If the equality holds, we shall call the graph X stable. Thus stabilitymeans Lft(p) = Aut(X). The octahedral graph O3 is an example ofan unstable graph: Let u, v be two non-adjacent vertices in O3. Thenα = (u1, v1) is an automorphism of Aut(X) interchanging the lifts u1

and v1 but fixing u0 and v0. Hence it does not preserve the fibresover u and v. One can prove that complete graphs are stable. On theother hand, the dodecahedral graph is unstable, as well as many othercubic arc-transitive graphs. Direct computation or checking the list byConder [CD02] shows that the dodecahedral graph is 2-transitive whileits canonical double cover is a 3-transitive cubic graph.

In case of imprimitive action of a group G on a graph X, grouptheorists use to form a (simple) quotient graph X whose vertices areblocks of imprimitivity, and they investigate the induced action of Gon X. Recall that the kernel K of the action is formed by elements ofG fixing each of the blocks set-wise is a normal subgroup K C G. Inparticular, CT(p) is a normal subgroup of the group of all lifts alonga covering p : X → X. Two blocks B1, B2 are adjacent if there is anedge uv in X, where u ∈ B1 and v ∈ B2. The mapping v 7→ Bv, where


v ∈ Bv, extends to a covering X → X if and only if the vertices of Bform an independent set.

The following lemma establishes the mentioned facts in case of reg-ular coverings explicitly.

Lemma 2.16. Let p : X → X be a regular covering between connectedgraphs and let A = Aut(X). Then the normalizer NAut(X)

(CT(p)) of

CT(p) projects. In particular, the normalizer NAut(X)(CT(p)) coin-

cides with the group A of all lifts.

Proof. Let A = CT (p). Then the covering p : X → X is equivalent toX → X/A; x 7→ [x]A. For each f ∈ NAut(X)

(CT(p)), the projection

f : [x]A → [fx]A is well-defined. Hence, all automorphisms in the nor-malizer NAut(X)

(CT(p)) are lifts of automorphisms in X. The inverseinclusion is clear. 2

Example 2.6. (Groups of fibre preserving automorphisms:) In a reg-ular covering p : X → X, the normalizer NAut(X)

(CT(p)) is the group

of all lifts of Aut(X). However, NAut(X)(CT(p)) may not coincide with

the group of fibre preserving automorphisms.For instance, take the complete bipartite graph Kn,n, n > 2, and

represent its vertices as Zn×Z2 so that V = j | j ∈ Zn∪j′ | j ∈ Znis the bipartition. Then one can decompose the edges of the completebipartite graph Kn,n into n prefect matchings M0,M1, . . . , Mn−1, whereMi is formed by the n edges joining a vertex j to a vertex (j + i)′ fori, j ∈ Zn. Coloring the edges of Mi by i ∈ Zn induces a regular coveringprojection p of Kn,n onto an n-valent dipole D(n) with CT(p) generatedby j 7→ j + 1 and j′ 7→ (j + 1)′ which is isomorphic to Zn.

Note the automorphism group G = Aut(D(n)) is isomorphic Sn×Z2

whose order is 2 · n!, hence

|Lft(Aut(X))| = |NAut(X)(CT(p))| = n|Aut(X)| = 2n · n!.

On the other hand, the automorphism group of Kn,n is Aut(Kn,n) ∼=Sn oZ2, the wreath product, which contains all permutations of verticesof each partite set and the interchanging of two partite sets and itswhose order is 2 · (n!)2. Moreover, all automorphisms of Kn,n are fibre


preserving and hence the group of fibre preserving automorphisms isthe full group Sn o Z2. Hence there are 2 · (n!)2 fibre preserving auto-morphisms and this number is grater than |NAut(X)

(CT(p))| = 2n · n!for n > 2.

Remark: The above results related with the semi-regularity of CT(p)can be alternatively explained from the point of view of Theorem 1.7by considering the monodromy action of the fundamental group πu onthe fibre fibu.

The next three results, much in the spirit of classical theory [Ma91],are aimed at a smooth transition to a combinatorial treatment of liftingautomorphisms. The first one states that, briefly speaking, an automor-phism of the base graph has a lift if and only if this automorphism isconsistent with the actions of the fundamental groups.

?f f

f

-

-

-

-

µ µ

µ µ

f

p

X

X

u

u

b

b

fu

f u

fb

f b

W

fW

p

W

?

p

?

p

?

p

?

fW

Figure 2.5: Proof of Theorem 2.17

Theorem 2.17. Let p : X → X be a covering projection of connectedgraphs and let f ∈ Aut(X). Then f lifts to an f ∈ Aut(X) if and only


if, for an arbitrarily chosen base vertex b ∈ X, there exists a bijectionφ : fibb → fibfb so that the pair

(φ, f∗) : (fibb, πb) → (fibfb, π

fb)

becomes an isomorphism of the fundamental group acting spaces andf |fibb

= φ.Moreover, there is a one-to-one correspondence f ↔ φ between

Lft(f) and all functions φ for which (φ, f∗) is such an isomorphism,with the relation

f u = φ(u ·W ) · fW−1 for any W : pu → b in X.

That is, f u is the terminal vertex of the lifting of the walk fW−1 whichis initiated at vertex φ(u ·W ).

Proof. (⇒) : Let f be a lift of f , and let vertices b ∈ X and u ∈ fibb begiven. We first show that for an arbitrary walk W : pu → b, we have

f(u ·W ) = f u · fW. (2.1)

Indeed, let W : u → u ·W be a lifting of W . Then f W : f u → f(u ·W )projects to fW , implying (2.1). In particular, (2.1) holds true for eachu ∈ fibb and W ∈ πb. Since φ := f |fibb

and f are bijections, we havethe required isomorphism of actions.

(⇐:) Conversely, let (φ, f∗) be such an isomorphism. Define therequired lift f on vertices of X as follows. Let u be an arbitrary vertexof X and let u = pu. Choose an arbitrary walk W : u → b and set

f u = φ(u ·W ) · fW−1.

We have to show several things. First of all, this mapping is welldefined, that is, it does not depend on the choice of W . Indeed, letW1, W2 : u → b be any two walks. Then u ·W1 = (u ·W2) ·W−1

2 W1.Therefore, φ(u ·W1) = φ(u ·W2) · f(W−1

2 W1) = φ(u ·W2) · fW−12 · fW1,

and hence φ(u ·W1) · fW−11 = φ(u ·W2) · fW−1

2 . Next, we see from thedefinition of f that

pf u = p(φ(u ·W ) · fW−1) = p(φ(u ·W )) · fW−1

= fb · fW−1 = f(b ·W−1) = fpu.


We verify that it is a bijection. To show that it is onto, let v be anarbitrary vertex of X and choose W : pv → fb arbitrarily. It is easilychecked that the vertex φ−1(v ·W ) · f−1W−1 is mapped to v. To showthe injectivity, let φ(u1 ·W1) · fW−1

1 = f u1 = f u2 = φ(u2 ·W2) · fW−12 .

Then pf(u1) = fpu1 = fpu2 = pf(u2) implying p(u1) = p(u2). Itfollows that u1, u2 are in the same fibre and by the first part of theproof we may assume W1 = W2 = W . Consequently, f u1 = f u2 impliesφ(u1 ·W ) · fW−1 = φ(u2 ·W ) · fW−1. Since φ is a bijection, it followsu1 = u2.

Having f defined on vertices it is immediate that the mapping ex-tends to darts. We conclude that f as above is a lift of f . This showsthat taking the restriction Lft(f) → Lft(f)|fibb

defines a function ontothe set of all such φ for which (φ, f∗) is an isomorphism of fundamentalgroup actions. The injectivity of this function follows from Proposi-tion 2.15. 2

It is a consequence of Theorem 2.17 that the problem of whether anautomorphism f has a lift can be reduced to the problem of whetherthere is an automorphism between the fibres fibb and fibfb considered asthe spaces of actions of fundamental groups. This in turn can be testedjust by comparing how the automorphism maps a stabilizer under theaction of the fundamental group. In this case, a stabilizer πb of the

action (fibb, πb) at b is the image of πb by the group homomorphism

p∗ : πb → πb. In addition, it enables us to give a more explicit formulaexpressing how an arbitrary lifted automorphism acts on the vertexfibres.

Theorem 2.18. Let p : X → X be a covering projection of connectedgraphs, and let f ∈ Aut(X) and b be a vertex of X. Then there existsan isomorphism of actions


fb)

if and only if f∗ maps the stabilizer πb of an arbitrarily chosen basepoint b ∈ fibb isomorphically onto some stabilizer πv ≤ πfb.

In this case we have v = φb. It gives a one-to-one correspondencev ↔ φ between the set v ∈ fibfb | f∗πb = πv and all such admissibleisomorphisms φ. In particular, the number of all liftings of f equals thecardinality |v ∈ fibfb | f∗πb = πv|.


Proof. (⇒) is clear.(⇐:) Let fπb

b= πfb

v . Each u ∈ fibb can be written as u = b · S for

some S ∈ πb because X is connected. Define φ by setting φu = v · fS.It is easy to check that (φ, f∗) is the required isomorphism of actions.The claimed one-to-one correspondence should also be clear since φ iscompletely determined by the image of one point. 2

As an immediate corollary of the above two theorems we have thefollowing observation. There exists a covering transformation mappingu1 ∈ fibu to u2 ∈ fibu if and only if the stabilizers coincide: πu1 = πu2 .From the fact that a covering projection is regular if and only if CT(p)acts transitively, a regular covering can be characterized by means ofstabilizers under the action of the fundamental group.

Corollary 2.19. A covering projection of connected graphs p : X → Xis regular if and only if, for an arbitrarily chosen vertex b of X, all thestabilizers under the action of πb on fibb coincide, that is to say, thestabilizer is a normal subgroup.

Recall that a stabilizer of the action (fibb, πb) at b is the image of

πb by the group homomorphism p∗ : πb → πb.

It follows from Lemma 2.16 that if a group of automorphisms Alifts along a regular covering projection p then it lifts to an extensionof CT(p) by A. Naturally, one can ask that under what condition thelifted group is isomorphic to a split extension CT(p)oA. This problemis investigated in [MNS00] and [Ma98]. It is a hard problem and willbe discussed later again. A special case is given as follows.

Proposition 2.20. Let p : X → X be a regular covering projection ofgraphs. Let A ≤ Aut(X)v be a subgroup of the stabilizer of a vertex v.If A lifts then its lifting is a split extension of CT(p).

Exercises

2.4.1. Determine the lift of f ∈ Aut(st3) of order 3 along the covering de-scribed in Example 2.1. Describe the group of lifts.

2.4.2. Prove that if X = Kn is the complete graph and p : X → X is thecanonical double covering then Lft(p) = Aut(X).


2.5 Lifting of graph automorphisms in terms

of voltages

A construction of a certain type of symmetrical graphs or maps is oneof interesting topics in algebraic or topological graph theory. In mostcases, it can be possible by using a lifting of automorphisms if thesymmetry is described in terms of an automorphism. One of classicalexamples is a construction of infinite families of cubic 5-arc-transitivegraphs given by D.Z. Djokovic, ”Automorphisms of graphs and cover-ings,” JCTB 16(1974), 243-247, or construction of highly symmetricalmaps introduced in Biggs [Bi74]. A lot of recently published papers(see for instance [Feng, Kwak, Marusic, Malnic]) employs a lifting ofautomorphism to construct several types of symmetric graphs. Theo-retical backgrounds were developed in a paper by Malnic, Nedela andSkoviera [MNS00].

Let p : X → X be a covering projection of connected graphs. ByTheorem 2.17, an automorphism f ∈ Aut(X) has a lift f if and onlyif, for an arbitrarily chosen base vertex b ∈ X, there exists a bijectionφ : fibb → fibfb so that the pair


fb)

becomes an isomorphism of the fundamental group acting spaces andf |fibb

= φ. Let (F, G; ξ) be a voltage space associated with the coveringp : X → X. As remarked in Section 2.2, the action of the fundamentalgroup πb on fibb is “the same, modulo relabelling” as the action of thelocal group Gb on F . Can we transfer the isomorphism problem betweenthe fibres as acting spaces of fundamental groups to a similar one byjust considering the local voltage groups? The answer is yes, providedthat certain additional requirements are imposed on the voltage space.This is especially relevant in the context of lifting groups rather thanindividual automorphisms.

Let A be a group of automorphisms of X. We say that the voltagespace (F,G; ξ) is locally A-invariant at a vertex v if, for every f ∈ Aand for every walk W ∈ πv, we have

ξW = 1 =⇒ ξfW = 1. (2.2)

2.5. Liftings: in terms of voltages 51

We remark that we can speak of local invariance without specifying thevertex, if a local invariance at a vertex implies the same at all vertices.A voltage space is locally f -invariant for an individual automorphismf of X if it is locally invariant with respect to the cyclic group 〈f〉generated by f . This is equivalent to requiring that condition (2.2) besatisfied by both f and f−1. If f has finite order or if X is a finitegraph, then it is even sufficient to check (2.2) for f alone.

Assume that the voltage space is locally A-invariant. Then for eachf ∈ A, its inverse f−1 must also satisfy condition (2.2). Hence, a localA-invariance is equivalent to the requirement that for each f ∈ A thereexists an induced isomorphism fu

# : Gu → Gfu of local voltage groupssuch that the diagram

πu πfu

Gu Gfu

-

-

fu∗

fu#

?ξ

?ξ

commutates; in other words, fu#(ξW ) = ξfuW for any W ∈ πu.

It is clear that in order to transfer the isomorphism problem be-tween the fibres as acting spaces of fundamental groups to local voltagegroups, we must require a local invariance in advance. If the voltagespace is, say, a monodromy one, then the existence of a lift implies alocal invariance. However, this needs not be true in general.

Theorem 2.21. Let (F, G; ξ) be a monodromy voltage space associ-ated with a covering p : X → X of connected graphs, and let f be anautomorphism of X. Then the following statements are equivalent:

(1) f has a lift,

(2) the voltage space (F, G; ξ) is locally f -invariant,

(3) the mapping f b# : Gb → Gfb taking ξW 7→ ξfW is a well-defined

isomorphism of the local voltage groups as acting groups on F .

Proof. (1) ⇒ (2). Let f have a lift f . By Theorem 2.17 the restric-tion φ = f |u gives an isomorphism (φ, f∗) of the actions (fibu, π

u) →(fibfu, π

fu). In particular, since (F,G; ξ) is an associated voltage space,


ξ(W ) = 1 implies that W ∈ πu acts on fibu as the identity. Since (φ, f∗)is an isomorphism of the actions, fW acts on fibfu as the identity. Since(F, G; ξ) is a monodromy voltage space, it follows that ξ(fW ) = 1 prov-ing (2).

To show (2) ⇒ (3), it is enough to show that fu#(ξW ) = ξfW for

W ∈ πu is well-defined. Let W1 and W2 be closed walks in πu suchthat ξW1 = ξW2 . We want to prove ξfW1 = ξfW2 . Then 1 = ξW1ξ

−1W2

=ξW1W−1

2∈ Gu. By the local f -invariance

1 = ξf(W1W−12 ) = ξf(W1)f(W−1

2 ) = ξf(W1)ξf(W−12 ) = ξf(W1)ξ

−1f(W2)

and we are done.(3) ⇒ (1). By Theorem 2.17, f has a lift if and only if f is consistent

with the actions of fundamental groups. We need to find a bijectionφ : fibu → fibfu such that φ(u · W ) = φ(u) · fW . Since (F, G; ξ) isan associated voltage space there are bijections ù : fibu → F and`fu : fibfu → F determining morphisms of actions (fibu, π

u) → (F,Gu),(fibfu, π

fu) → (F,Gfu), respectively. By assumption, fu# exists and it

gives an isomorphism of actions (ϕ, fu#) : (F,Gu) → (F,Gfu). We set

φ = `−1fuϕù. Since fu

# is consistent with the actions of Gu and Gfu, fis consistent with the actions of the respective fundamental groups onthe fibres. That is, the diagram

(fibu, πu) (fibfu, π

fu)

(F,Gu) (F,Gfu)

-

-

(φ, fu∗ )

(ϕ, fu#)?

(ù, ξ)

?

(`fu, ξ)

commutates. 2

For a regular voltage space associated with a regular covering, wecan immediately infer that an automorphism lifts if and only if everyclosed walk of the trivial voltage is mapped to a closed walk of the


same voltage. This result was previously observed in context of liftingautomorphisms of regular maps in [GS93], [NS97] and in context ofenumeration of coverings in [HKL96, HKL96].

Corollary 2.22. Let the covering as well as the voltage space be regular.Then the following three statements are equivalent:

(1) f has a lift,

(2) the voltage space (F, G; ξ) is locally f -invariant,

(3) the mapping f b# : Gb → Gfb taking ξW 7→ ξfW is a well-defined

isomorphism of the local voltage groups with respective actions onF .

Moreover, if (G,G, ξ) is a Cayley voltage space then Gb = G = Gfb andf b

# is an automorphism of G.

Note that in Condition (3) in Corollary 2.22, we could erase therequirement on consistency of the isomorphism ξW 7→ ξfW , since byregularity any group isomorphism is consistent with the (right) actionof G on F . In particular, if (G,G, ξ) is a Cayley voltage space thengroup automorphisms are by definition consistent with the action of Gon itself given by right multiplication. In what follows we show that itdoes not hold for locally transitive monodromy voltage spaces.

Example 2.7. Let X be a two-vertex graph with a singe edge joiningthem and two loops at each vertex. Formally let u, v be vertices andlet D+ = xu, yu, xv, yv, z be an orientation of X, I(xu) = I(yu) =I(x−1

u ) = I(y−1u ) = u, I(xv) = I(yv) = I(x−1

v ) = I(y−1v ) = v. Define a

permutation voltage space on X by setting F = 1, 2, 3, 4, ξxu = (1, 2),ξyu = (3, 4), ξxv = (1, 3)(2, 4), ξxu = (1, 4)(2, 3) and ξz = 1. It is easy tosee that the local groups Gu = Gv = 〈(1, 2), (1, 3)(2, 4)〉 is the dihedralgroup of order 8. The mapping f : xu 7→ yv, yu 7→ xv, z 7→ z−1 extendsto a graph automorphism interchanging the vertices u, v. Then f# takes(1, 2) 7→ (1, 4)(2, 3) and (1, 3)(2, 4) 7→ (3, 4). Since it maps generators〈(1, 2), (1, 3)(2, 4)〉 onto generators 〈(1, 4)(2, 3), (3, 4)〉 and the definingrelations are preserved, it extends to a group automorphism of D4 whichis clearly not consistent with the action. Actually, it can be seen directly(inspecting the derived graph) that f has no lift.


The next corollary constitutes the last stage of combinatorializationof the lifting problem by reducing it to a system of equations in thesymmetric group Sn, where n is the size of the fibre.

Corollary 2.23. Let (F, SymR F ; ξ) be a locally transitive permutationvoltage space on a graph X. Then, an automorphism f of X has a liftif and only if the system of equations

ξS · τ = τ · ξfS

in SymR F has a solution τ , where S runs through a generating set forπb.

Moreover, there is a one-to-one correspondence between all solutionsand the lifts of f .

Proof. By Theorem 2.21(3) the mapping f b# : ξW 7→ ξf#W , W ∈ πb,

is a group isomorphism Gb → Gfb consistent with the permutationactions. This happens if and only if there is a permutation ϕ : F → Fsuch that ϕ(x · g) = ϕ(x) · f b

#(g) ∀g ∈ Gb. Representing the bijection

ϕ as an element τ of SymR F and taking g = ξW for some W ∈ πb

we get ξW · τ = τ · ξf#W . In particular, it holds for W forming thefundamental cycles with respect to some spanning tree. Conversely,having the solution of ξS · τ = τ · ξfS, for the generators of Gu theconsistent isomorphism fu

# is determined by the images of generatorsfu

#(S) = ξfS = τ−1ξSτ .To see the last part of the statement assume that 1 · ξS = i for some

element S of πu. Let j = 1 · ξfS = 1 · τ−1ξSτ . We have f(ui) = vj foreach ui ∈ fibu, i = 1, 2, . . . , n. This way f is determined on fibu. Thenby Theorem 2.21 f is determined everywhere. Thus a solution τ of theabove system of identities determines precisely one lift. 2

Example 2.8. (Connected regular covers of the semistar st3 on whichAut(st3) can lift:) Take the semistar st3 as the base graph and denotea, b, c the three semiedges. Clearly, Aut(st3) = Sym(a, b, c) is the sym-metric group of degree 3, which is generated by two cycles f = (a b c)and g = (b c). The fundamental group πu = 〈a, b, c | a2 = b2 =c2〉 = Z2 ∗Z2 ∗Z2. The system of equations for lifting Aut(st3) follows:ξa ·τ = τ ·ξb, ξb ·τ = τ ·ξc, ξc ·τ = τ ·ξa for lifting f , and ξa ·ω = ω ·ξa,ξb · ω = ω · ξc, ξc · ω = ω · ξb for lifting g.


Indeed, a connected regular cover of the semistar st3 on whichAut(st3) can lift is isomorphic to a cubic Cayley graph based on a groupG generated by 3-involutions and admitting group automorphisms per-muting the generators following the action of Sym(a, b, c). (See Exam-ple 2.3).

The least examples include K4, K3,3, Q3 and GP (8, 3). However,the Petersen graph does not cover st3 at all, since such a covering wouldproduce a 3-edge-colouring.

Example 2.9. (All n-fold coverings of the Petersen graph P on whichAut(P ) lifts:) Let V (P ) = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 being the vertex set

0 1

2

34

5

6

7

89

Figure 2.6: The Petersen graph.

labelled as in Figure 2.6. Since every covering projection of the Petersengraph P can be described by means of a permutation voltage space, itis sufficient to characterize all the permutation voltage assignments onthe darts of P such that the derived covering has the required property.By Proposition 2.9 we may assume that the darts of an arbitrarilychosen spanning tree carry the identity permutation. Consequently,every covering projection P → P is completely described by a 6-tupleof permutations ξ = (ξ0, ξ1, . . . , ξ5), where ξi is the voltage carried bythe dart xi joining i to i + 1, for i = 0, 1, . . . , 5 with arithmetic mod 6.Since Aut(P ) is generated by the automorphisms

f = (012345)(876), g = (03)(12)(45)(78), h = (01845)(29376),


Corollary 2.23 gives us three systems of equations ξSτ = τξfS, ξSη =ηξgS and ξSω = ωξhS in Sn, where S ∈ π9 ranges over all elementsof the fundamental group at the vertex 9. The fundamental group π9

is generated by the following six closed walks (for brevity indicatedjust as cyclic sequences of vertices) W0 = (96018), W1 = (98127),W2 = (97236), W3 = (96348), W4 = (98457) and W5 = (97506). Thusthe above system of equations reduce to the following 18 equations invariables ξ0, . . . , ξ5, τ , η, and ω, where every generator of π9 and everygenerator of Aut(P ) gives rise to one equation:

ξ0τ = τξ1 ξ0ω = ωξ−11 ξ0η = ηξ−1

2 ξ−11 ξ−1

3

ξ1τ = τξ2 ξ1ω = ωξ−10 ξ1η = ηξ3

ξ2τ = τξ3 ξ5ω = ωξ−12 ξ2η = ηξ2

ξ3τ = τξ4 ξ2ω = ωξ−15 ξ3η = ηξ−1

2 ξ−14 ξ−1

3

ξ4τ = τξ5 ξ4ω = ωξ−13 ξ4η = ηξ3ξ4ξ5

ξ5τ = τξ0 ξ3ω = ωξ−14 ξ5η = ηξ0ξ1ξ2.

From the first column it easily follows that the required six voltagesξi can be expressed as ξi = τ−iντ i, i = 0, 1, . . . , 5, where ν and τ arepermutations satisfying the identity ντ 6 = τ 6ν. Thus f lifts if and onlyif the covering projection is determined by the 6-tuple

(ξ0, ξ1, . . . , ξ5) = (ν, τ−1ντ, τ−2ντ 2, τ−3ντ 3, τ−4ντ 4, τ−5ντ 5),

where ν and τ satisfy the identity ντ 6 = τ 6ν. For instance, any choice ofν combined with any choice of τ of order 2, 3 or 6 is good. By specifyingτ we actually prescribe how f should lift. For the automorphism groupAut(P ) = 〈f, g, h〉 we get: the group Aut(P ) lifts if and only if thecover of P is determined by the 6-tuple of voltages

(ν, τ−1ντ, τ−2ντ 2, τ−3ντ 3, τ−4ντ 4, τ−5ντ 5),

where ν and τ are two of the four generators of G = 〈ν, τ, η, ω; (R), . . . 〉and (R) is the system of 13 identities arising from the above system ofequations if we replace the first column by the identity ντ 6 = τ 6ν andsubstitute all ξi by τ−iντ i in all the other equations.

The above permutation group G = 〈ν, τ, η, ω; (R), . . . 〉 can be viewedas an image of a universal 4-generator group G generated by four ab-stract generators ν, τ, η, ω and satisfying the set of identities (R). Each


‘admissible voltage group’ for the investigated lifting problem is a quo-tient of G. If the abelianization of G is infinite then there are infinitelymany solutions of the considered lifting problem.

Example 2.10. (Continuation of Example 2.9) By taking ξi = (12) ∈S2 for i = 0, 1, . . . , 5, we get the generalized Petersen graph GP (10, 3)as a cover of the Petersen graph such that the full automorphism grouplifts. The solutions of the above system of equations are τ = ω = η = 1in S2.

If we set ξ0 = ξ2 = ξ4 = (12)(34), ξ1 = ξ3 = ξ5 = (13)(24) in thegroup S4, another interesting example is obtained. In this case, thesolutions of the above system of equations are τ = (14), ω = (1243)and η = (14)(23). Thus the derived covering graph is a 3-arc-transitivecubic graph and has 40 vertices. In fact, the resulting graph can beviewed as a canonical double covering D of the dodecahedron D. Since|Aut D| > |Aut D × Z2| it follows that the dodecahedron is unstable(see [MS93] for the definition and [NS96] for further information andresults). This is the first symmetrical example of its kind, as D issharply 2-arc-transitive.

Example 2.11. (Irregular covers of Kn on which Aut(Kn) lifts:) Letthe vertex set of complete graph X = Kn be V = 0, 1, 2, . . . , n − 1and dart-set be D = (V × V ) \ (j, j)|j ∈ V as expected. Choose thespanning tree T of Kn to be the star containing edges (0, j), (j, 0)for j = 1, 2, . . . , n − 1. We define a T -reduced permutation voltagespace (F, H, ξ) on X, where F = Zn − 0 and H ⊆ Sym(F ), bysetting ξ(i,j) = (i, j) for each cotree dart (i, j). To show that ξ is locallyG = Aut(X) invariant observe that G = Sym(V ) is generated by alltranspositions (i, j), where i, j ∈ F , together with a single transposition(0, 1). Given transposition (i, j) (i, j ∈ F ) the system of equations fromCorollary 2.23

(i, j) · τ(i,j) = τ(i,j) · (i, j)(i, k) · τ(i,j) = τ(i,j) · (j, k) for all k 6= 0(j, k) · τ(i,j) = τ(i,j) · (i, k) for all k 6= 0(k, `) · τ(i,j) = τ(i,j) · (k, `) for all k, ` 6= 0


has a solution τ(i,j) = (i, j).Similarly for an automorphism (0, 1) ∈ Aut(X) the associated sys-

tem of equations

(k, 1) · τ(0,1) = τ(i,j) · (k, 1) for all k 6= 0, 1(i, j) · τ(0,1) = τ(0,1) · (i, j) for all i, j 6= 0, 1

has a trivial solution τ(0,1) = 1. Hence by Corollary 2.23 the automor-phism group of Kn lifts. The action of the permutation voltage groupon F is obviously irregular since the voltage assignments are not regularpermutations. In fact, the covering transformation group CT(p) = 1 istrivial. One can see it as follows. The voltage group H is generatedby all the transpositions in Sym(F ) hence H = Sym(F ) acts transi-tively on F with the point stabilizer isomorphic of S(1) ∼= Sn−2. SinceCT(p) is given by the action on a single fibre and by Theorem 1.7CT(p) ∼= NH(S(1))/S(1). However, conjugation by any permutationoutside S1 does not preserve S1 thus NH(S(1)) = S(1) and CT(p) = 1.On the other hand, the covering graphs are vertex-transitive. One canconstruct them as first subdividing each edge by introducing a vertexof degree two and then taking the line graph of the subdivision of Kn.

Exercises

2.5.1. With the above notation prove that the following permutation voltageassignment in S3 defined on K5 is locally invariant: ξ(1, 2) = ξ(2, 1) =ξ(3, 4) = ξ(4, 3) = (1, 2), ξ(1, 3) = ξ(3, 1) = ξ(2, 4) = ξ(4, 2) = (1, 3)and ξ(1, 4) = ξ(4, 1) = ξ(2, 3) = ξ(3, 2) = (2, 3). Compute CT(p).

2.5.2. Show that in general the existence of a lift of an automorphism doesnot imply the local invariance.

2.5.3. The Heawood graph is the point-line incidence graph of the Fano plane.Is the Heawood graph a regular cover over st3. If yes, describe the as-sociated Cayley voltage space. Can one construct the Heawood graphfrom a Aut(st3)-locally invariant voltage space over st3?

2.5.4. Prove that the lifted graph X in the Example 3.1 is vertex-transitive.2.5.5. Using the lifting condition prove that if k2 ≡ ±1(mod n) then the

generalised Petersen graph GP (n, k) is vertex-transitive.2.5.6. For all n ≤ 5 describe n-folded coverings of K4 such that the whole

automorphism group lifts.

Lifting problem, case of abelian CT(p) 59

1

2

3 4

6

5

id

idid

id id

x2

x1x3

x4

-R

µ

¾

Figure 2.7: K3,3 with T -reduced voltages

2.5.7. Find a solution of the lifting problem in terms of group presentations(as simple as possible) in case A = AutX and X ranges through theone-skeletons of the five Platonic solids.

2.6 Lifting problem, case of abelian CT(p)

Let X be a connected graph and let T be a spanning tree of X. Let

D+ = x1, . . . , xβ, xβ+1, . . . , xβ+tbe an orientation of X, where x1, . . . , xβ are the cotree darts andxβ+1, . . . , xβ+t are the tree darts, so that β + t = |D+|. Any automor-phism f of X can be represented by a β × (β + t) matrix C = CT (f),whose rows correspond to the β cotree darts in D+ and columns cor-respond to the darts in D+. Any cotree dart xi determines a uniqueelementary cycle Ci in T +xi which is directed consistently with xi. LetfCi = (x

εi,1

1 , xεi,2

2 , . . . , xεi,β+t

β+t ) be an image of the cycle Ci considered asa β + t vector. The exponent εi,j is 1 if xi appears in fCi, it is −1if x−1

i appears in fCi and it is 0 otherwise. The entries of CT (f) arethe exponents εi,j. The associated reduced matrix MT (f) is a β × βmatrix formed by the first β columns of CT (f) corresponding to thecotree darts.

Example 2.12. Let X = K3,3 be the complete bipartite graph withthe bipartition V = 1, 3, 5 ∪ 2, 4, 6. Choose the spanning tree T


to be the path T = 123456 as illustrated by dark lines in Figure 2.7.We orient the cotree darts by setting x1 = 14, x2 = 25, x3 = 36and x4 = 61. The corresponding elementary cycles, defined as cyclicpermutations of vertices, are: C1 = (1432), C2 = (2543), C3 = (3654)and C4 = (123456). (They are the generators of the homology groupH1(X).) Let r and ` be the automorphisms of X given by permutationsof vertices as follows: r = (1 2 3 4 5 6), ` = (2 4 6) as cycles. We haver(C1) = (2543), r(C2) = (3654), r(C3) = (4165) and r(C4) = (123456).Similarly, `(C1) = (1634), `(C2) = (4563), `(C3) = (3256) and `(C4) =(143652). One easily gets

MT (r) =

0 1 0 00 0 1 0

−1 0 0 −10 0 0 1

MT (`) =

−1 0 −1 −10 0 −1 00 1 −1 01 −1 1 0

.

Note that the i-th rows of MT (r) and MT (`) represent the elementsr(Ci) and `(Ci), respectively in the homology group H1(X).

A square matrix is called unimodular if its determinant is ±1. Theset of n×n unimodular integer matrices forms a group under the matrixmultiplication, called the unimodular (matrix) group and denoted bySL±(n,Z).

Proposition 2.24. Let X be a connected graph with a spanning treeT . Then the mapping f 7→ MT (f) is a homomorphism from Aut(X)to the unimodular group SL±(β,Z), defining a matrix representation ofAut(X).

Proof. For graphs without semiedges the proof is done in [Sir01]. Weextend it onto graphs with semiedges. Let X be a graph, f ∈ Aut(X)and let MT (f) be a matrix representation of f with respect to a span-ning tree T and an orientation D+. First, we observe that

(1) a semiedge is always a cotree edge, that is, a semiedge does notbelong to any spanning tree T ,

(2) each semiedge x determines an elementary cycle Cj(x) orientedconsistently with the orientation of x,

2.6. Liftings: case of abelian CT(p) 61

(3) the image of a semiedge under a graph automorphism f is againa semiedge.

By (1) and (2) we can reorder the cotree edges and elementary cyclesof X such that the first s rows and the first s columns correspond tothe set of semiedges of X. Recall that a transposition of two rows (orcolumns) of a matrix preserves the determinant up to the sign ±1. Thecondition (3) implies that the matrix M = MT (f) can be then expressed

as a block matrix

[S 00 C

], where S is an s × s permutation matrix

representing the action of f on the semiedges, C is (β − s) × (β − s)matrix representing action of f on elementary cycles induced by thelinks and loops. Since the matrix C represents the action of f ona subgraph X ′ ⊆ X induced by the sets of links and loops of X, itis unimodular. The matrix S is a permutation matrix, and hence it isunimodular. Since both C and S are also unimodular, M is unimodularas well. 2

Given an abelian group A, the Cartesian product Aβ can be con-sidered as a β-dimensional Z-module. If K is a submodule then theset of integer (column) vectors ~a = (a1, a2, . . . , aβ) ∈ Zβ such thata1x1 + a2x2 + · · ·+ aβxβ = 0 ∈ A for each ~x = (x1, x2, . . . , xβ) ∈ K willbe called an orthogonal complement of K and will be denoted K⊥. IfK = 〈x〉 we write x⊥ instead of 〈x〉⊥.

If (A,A, ξ) is a Cayley voltage space on X reduced with respect toT , we denote by

~ξ = (ξC1 , ξC2 , . . . , ξCβ) =

ξC1

ξC2

...ξCβ

β×1

the β dimensional column vector of voltages on the elementary cy-cles. Call ~ξ the matrix representation of a voltage ~ξ with respect toa spanning tree T . Note that for each Ci we may form a uniquegenerator Si = WiCiW

−1i in the fundamental group πu, where Wi

is the unique path in T (possibly trivial) joining u to Ci. Since ξis T -reduced, ξSi

= ξCifor all i = 1, . . . , β. Since A is abelian,


ξfSi= ξfCi

for all i = 1, . . . , β and any automorphism f . Note that

MT (f)~ξ = (ξf#C1 , ξf#C2 , . . . , ξf#Cβ). Since the local voltage group is

Au = ξ(πu) = ∑i aiξSi| ai ∈ Z, it follows that

ξW = 0 for W =∏

i aiSi if and only if ~a = (a1, a2, . . . , aβ) ∈ ~ξ⊥,

the orthogonal complement of ~ξ. Moreover, since A is abelian for anyW in π(X) there is W ′ =

∏i aiSi such that ξW = ξW ′ =

∑i aiξSi

=∑i aiξCi

.

Theorem 2.25. Let X be a connected graph with cycle rank β anda spanning tree T and let (A,A, ξ) be a T -reduced locally transitiveCayley voltage space on X with an abelian group A. Then the followingstatements are equivalent.

(1) An automorphism f of X lifts,

(2) (A,A, ξ) is locally f -invariant,

(3) ~ξ⊥ = (M~ξ)⊥, that is, the orthogonal complements of the vectors~ξ and M~ξ are identical, where M = MT (f) is the matrix repre-sentation of f with respect to the spanning tree T .

Moreover, if A = Zn is cyclic then the above statements are equivalentto

(4) there exists an eigenvalue α coprime to n such that M~ξ = α~ξ.

Proof. By Theorem 2.21 (1) ⇔ (2).

(2) ⇒ (3) If (2) holds then by Theorem 2.21(3) the automorphismf induces an isomorphism fu

# of the local groups Au → Afu given byξW 7→ ξfW . Since the voltage space is T -reduced and locally transitive,it follows Au = A for each u ∈ V . Thus fu

# = f# : A → A is a group

automorphism. Let 0 = ~a · ~ξ =∑β

i=1 aiξi, where ~a · ~ξ means the dotproduct. Then

0 = f#

(β∑

i=1

aiξi

)=

β∑i=1

aif#ξi =

β∑i=1

aif#ξSi=

β∑i=1

aiξfSi= ~a ·M~ξ.


It shows (~ξ)⊥ ⊆ (M~ξ)⊥. To prove (M~ξ)⊥ ⊆ (~ξ)⊥ we do a similar

calculation. Assuming 0 = ~a ·M~ξ =∑β

i=1 aiξfSiand using (f#)−1 we

have

0 = (f#)−1

(β∑

i=1

aiξf(Si)

)=

β∑i=1

ai(f#)−1ξf(Si)

=

β∑i=1

ai(f−1)#ξf(Si) =

β∑i=1

aiξSi= ~a · ~ξ. (2.3)

(3) ⇒ (2) Suppose that the orthogonal complements are identical.Assume ξW = 0 for an u-based closed cycle W . Since W is expressible interms of the β generators S1, S2, . . . , Sβ and since A is abelian, we have

0 = ξW =∑β

i=1 aiξSi= ~a · ~ξ for some integer vector ~a = (a1, a2, . . . , aβ),

where ai = bi − ci for i = 1, . . . , β and bi, ci are the number of timesthe cotree darts xi, x−1

i , respectively, appears in W . By assumption

0 = ~a · M~ξ =∑β

i=1 aiξfSi= ξfW . It follows that (A,A, ξ) is locally

f -invariant.To show the last statement, let A ∼= Zn. Assume any of the con-

ditions (1), (2) and (3) holds. Since every group automorphism of thecyclic group Zn is of the form h 7→ αh for some α coprime to n, wehave M~ξ = f#

~ξ = α~ξ. Conversely, if M~ξ = α~ξ, it holds (~ξ)⊥ = (M~ξ)⊥

clearly showing (3). 2

Example 2.13. (The canonical double covering of a nonbipartite graph)For every nonbipartite graph X one can easily define a Cayley voltagespace (Z2,Z2, ξ) which is Aut(X)-locally invariant by setting ξx = 1 forevery dart x. Indeed, ξW = 0 if and only if |W | is even. If |W | is eventhen |f#W | is even for any f ∈ Aut(X). The existence of an odd cycleimplies that (Z2,Z2, ξ) is locally transitive, and so the derived graphX is connected. The covering graph X is called the canonical doublecovering of X [NS96]. It is a minimal bipartite cover over X in thefollowing sense: If Y is a bipartite cover of X then it covers X.

Example 2.14. (Infinitely many connected coverings over X on whichAut(X) can lift:) Let X be a connected graph. Choose an arbitrary


spanning tree T of X and let n > 0 be the number of cotree links andloops, and let m be the number of semiedges. Set A = Zn

k ⊕ Zm2 for

any positive integer k > 1 and n > 0. With a preferred orientation onX, define a voltage ξ in such a way that the darts of the chosen span-ning tree receive the trivial voltage, the voltages on darts of semiedgesgenerate the Z2-part of A, and the voltages on darts of links or loopsof cotree edges generate the Zk-part of A. However, the Z-modules Zn

k

and Zm2 are special ones. In these Z-modules, any two generating sets of

size n, m respectively, have the same orthogonal complement consistingof integer vectors whose entries are divisible by k or 2, respectively.

Since every f ∈ Aut(X) maps the links, loops and semiedges to

links, loops and semiedges respectively, it follows that the vectors ~ξ =(ξC1 , ξC2 , . . . , ξCβ

) and MT (f)~ξ = (ξf#C1 , ξf#C2 , . . . , ξf#Cβ) have the same

orthogonal complement. Hence, by Theorem 2.25(3) Aut(X) lifts.Since n > 0, one can form an infinite family of regular covers overX by taking the free parameter k to be any integer. Another infinitefamily of coverings p : X → X such that Aut(X) lifts along p can beobtained by iterating the above construction.

We summarize the above construction as follows.

Proposition 2.26. Let X be a connected graph with at least one cycle.Then there are infinitely many finite connected regular covers over Xon which Aut(X) lifts.

Remark: The construction in Example 2.14 generalizes to the caseinvolving an arbitrary coefficient ring, thereby extending homologicalcoverings of Biggs [Bi74, Bi84] and Surowski [Su88] to graphs withsemiedges. The name homological coverings come from the followingconsideration. Given a connected graph X and its spanning tree Twe may form a T -reduced voltage assignment ξ : D → πu(X) takingvalues in the fundamental group by setting trivial voltages on the treedarts and taking the respective generator of the fundamental groupinduced by x for any cotree dart x. The reader might observed thatthe covering derived from this Cayley voltage assignment is a universalcovering. Replacing the fundamental group by its abelianisation weget another (homological) voltage assignment. However, the universal


abelian cover may still be infinite. Factoring the infinite cyclic factorsmodulo a chosen integer k, we get a homological cover reduced modulok which is finite for every k.

Example 2.15. (All cyclic coverings of K3,3 on which an automor-phism group G = 〈r, `〉 lifts:) With the notation introduced in Exam-ple 2.12 we want to describe all Cayley voltage spaces (Zn,Zn, ξ) onX = K3,3 such that the group of automorphisms G = 〈r, `〉 lifts, withr = (1 2 3 4 5 6), ` = (2 4 6). Let R := MT (r) and L := MT (`) bethe matrix representations of r and `, respectively derived in Exam-ple 2.12. By Theorem 2.25 we need to solve the system of congruencesR~ξ = α~ξ, L~ξ = β~ξ in variables ξ1, ξ2, ξ3, ξ4, α, β modulo n, where α,β are coprimes to n. First four congruences from R~ξ = α~ξ have asolution ~ξ = c(1, α, α2,−α3 − 1), where the parameters are related byc(α3 + 1)(α− 1) ≡ 0(mod n). Since we require (Zn,Zn, ξ) to be locallytransitive, c is coprime to n. It follows that we may assume c = 1.

The system L~ξ = β~ξ consists of congruences −(ξ1 + ξ3 + ξ4) = βξ1,−ξ3 = βξ2, ξ2 − ξ3 = βξ3 and ξ1 − ξ2 + ξ3 = βξ4. Inserting ξ2 = αand ξ3 = α2 in the second equation we get β = −α which allowsus to get rid of the parameter β. The third equation is equivalentwith α2 − α + 1 ≡ 0(mod n). The fourth equation translates intoξ4 = (−α)(α2 − α + 1) ≡ 0(mod n). The first equation reduces to thesame congruence α2 − α + 1 ≡ 0(mod n) as the second one. Observethat multiplying both sides of α2 − α + 1 ≡ 0 by α2 − 1 we get thecongruence (α3+1)(α−1) ≡ 0, required for L(r)~ξ = α~ξ. We summarizethe above computation as follows.

The group G = 〈r, `〉 lifts if and only if there is α in Zn such thatα is coprime to n and α2 − α + 1 ≡ 0(mod n). In such case the T -reduced voltage assignment is (up to a multiplicative constant) defined

by ~ξ = (1, α, α2, 0).Few small solutions of the congruence are α = 1, 2, 3, 4 when n =

2, 3, 7, 13, respectively. The group G = 〈r, `〉 ≤ Aut(K3,3) defined aboveis transitive on darts and it is the orientation preserving automorphismgroup of a hexagonal embedding of K3,3 in the torus. It follows thatthe lifts are arc-transitive cubic graphs. The above family is describedin [FK03] in another way.


Example 2.16. (Cyclic coverings of a dipole on which 〈ρ, λ〉 lifts:)Recall that a k-dipole is a graph whose dart-set is D = Zk × Z2, thevertex set is V = Z2, the incidence function is the projection on thesecond coordinate and the dart-reversing involution takes (i, j) 7→ (i, j+1). Fix orientation as (i, j) 7→ (i, j +1). Denote by ρ the automorphismD → D taking (i, j) 7→ (i + 1, j) and by λ the automorphism taking(i, j) 7→ (−i, j+1). Obviously ρ is a cyclic permutation of darts of orderk fixing both vertices, while λ can be viewed as a 180 degree rotationround an axis fixing the central point of the edge (0, 0), (0, 1). Wechoose a spanning tree T such that it contains the unique tree edge(0, 0), (0, 1).

The matrices representing the automorphisms ρ and λ with respectto T are obtained as

MT (ρ) =

−1 1 0 0 . . . 0 0−1 0 1 0 . . . 0 0−1 0 0 1 . . . 0 0...

......

... · · · ......

−1 0 0 0 . . . 1 0−1 0 0 0 . . . 0 1−1 0 0 0 . . . 0 0

(k−1)×(k−1)

(2.4)

MT (λ) =

0 0 0 . . . 0 0 −10 0 0 . . . 0 −1 00 0 0 . . . −1 0 0...

...... · · · ...

......

0 0 −1 . . . 0 0 00 −1 0 . . . 0 0 0

−1 0 0 . . . 0 0 0

(k−1)×(k−1)

. (2.5)

We want to describe all the locally transitive T -reduced Cayley volt-age spaces (Zn,Zn, ξ) such that the group of automorphisms G = 〈ρ, τ〉lifts. Let ξ = (ξ1, ξ2, . . . , ξk−1) be the vector of voltages in such avoltage space, more precisely ξ(i, 0) = ξi, for i = 1, 2, . . . , k − 1. ByTheorem 2.25, ρ lifts if and only if there exists α ∈ Z∗n such that


MT (ρ)~ξ = α~ξ. We write it as a system of equations in Zn:

−ξ1 + ξ2 = αξ1

−ξ1 + ξ3 = αξ2

−ξ1 + ξ4 = αξ3

. . .−ξ1 + ξk−1 = αξk−2

−ξ1 = αξk−1

This system has a solution: ξj = ξ1(1 + α + α2 + · · · + αj−1) forj = 1, 2, . . . , k − 1, where ξ1 and α are parameters, α is coprime ton satisfying

ξ1(1 + α + α2 + · · ·+ αk−1) = 0 (mod n).

Since we assume that our voltage space is locally transitive, we may setξ1 = 1. We summarize it as follows:

The automorphism ρ : D → D taking (i, j) 7→ (i+1, j) on the dipole liftsif and only if the T -reduced voltage-assignment is (up to multiplicativeconstant) defined by

ξj = 1 + α + α2 + · · ·+ αj−1 for j = 1, 2, . . . , k − 1,

where α is coprime to n satisfying

1 + α + α2 + · · ·+ αk−1 = 0 (mod n).

Again, by Theorem 2.25, λ lifts if and only if there exists β ∈ Z∗nsuch that MT (λ)~ξ = β~ξ. Rewriting it as a system of equations in Zn,we get

−(1 + α + α2 + · · ·+ αk−j) = β(1 + α + α2 + · · ·+ αj−1),

for j = 1, 2, . . . , k− 1. In particular, for j = 1 and for j = k− 1 we get

−(1 + α + α2 + · · ·+ αk−2) = β,

and−1 = β(1 + α + α2 + · · ·+ αk−2).


Thus β2 = 1. Similarly, inserting from the second equation (j = 2) intothe first one (j = 1) we get β(1 + α)− αk−2 = β. Using β2 = 1 and αis coprime to n, we derive αk−3 = β. Inserting third equation into thefirst one we deduce

β(1 + α + α2)− αk−3 − αk−2 = β.

This gives

β(1 + α + α2)− β − αβ = β.

Finally we derive α2 = 1.Assume first that k is odd. By αk−3 = β we get β = 1. But if β = 1

our system of equations reduces to just one equation 1+α +α2 + · · ·+αk−2 = −1, which by α2 = 1 gives (1 + α)k−1

2= −1. Multiplying by α

we get (1 + α)k−12

= −α which implies α = 1. Hence we have a uniquesolution ξi = i for i = 1, 2, . . . , k − 1.

Assume k is even. By αk−3 = β we get α = β. Our system ofequations reduces to 1 + α + α2 + · · · + αk−2 = −α, which hold truefor α satisfying α2 = 1. Finally, 1 + α + α2 + · · · + αk−1 = 0 (mod n)reduces to (1 + α)k

2= 0 (mod n).

We summarize the above computations as follows.

G = 〈ρ, λ〉 lifts if and only if the T -reduced voltage-assignment ξ (upto multiplicative constant) satisfies the following conditions:

(1) If k is odd then k = 0(mod n) and ξj = j for j = 1, 2, . . . , k − 1,

(2) if k is even then there exists α such that α2 = 1(mod n) and(1 + α)k

2= 0 (mod n), for each such an α there is a locally G-

invariant voltage assignment defined by ξ2j = j(1 + α) and byξ2j−1 = (j − 1)(1 + α) + 1.

It follows that a k-dipole of odd valency admits (up to multiplicationin Z∗n) exactly one T -reduced voltage assignment in a cyclic group Zn.On the other hand, if k is even there are at least two essentially differentadmissible voltages corresponding to α = ±1. The group G = 〈r, `〉 isdihedral and it is the group of orientation preserving automorphismsof the spherical embedding of k-dipole. Note that if α = 1 and k = nthen the derived graph is Kn,n.

Lifting problem, case of elementary abelian CT(p) 69

Exercises

2.6.1. Prove that K4 has only one non-trivial regular cover with cyclic CT(p)such that the full automorphism group lifts.

2.6.2. Prove that K3,3 has a non-trivial regular cover with cyclic CT(p) ∼= Zn

such that the full automorphism lifts if and only if n = 3. Describethe associated Cayley voltage space.

2.6.3. * Is there a regular cover of K3,3 with abelian CT(p) such that thefull automorphism group lifts and CT(p) is generated by two or threegenerators?

2.6.4. * Prove that there are infinitely many 5-transitive cubic graphs.

Hint: Let a vertex set V be the set of all involutory permutations inS6 which are either transpositions or can be expressed as a productof three disjoint transpositions. Two such permutations of differentcycle structure are joined by a link if they commute. Show that theabove defined graph X is a cubic 5-transitive graph.

2.6.5. * Describe all the cyclic covers over the octahedral graph and overthe cube Q3 such that the full automorphism group lifts.

2.6.6. * Employing the computation in Example 2.9 describe all the cycliccovers over the Petersen graph such that the full automorphism grouplifts.

2.6.7. * Prove Proposition 2.24.2.6.8. * A totally unimodular matrix is an integer matrix for which every

square non-singular submatrix is also unimodular. Prove: For a matrixM , if

(1) all its entries are either 0,−1 or +1;(2) any column has at most two nonzero entries; and(3) the column with two nonzero entries have entries with opposite

sign,

then the matrix M is totally unimodular (but the converse is not true).2.6.9. Prove that any permutation matrix is totally unimodular.

2.7 Lifting problem, case of elementary

abelian CT(p)

Let X be a graph and let f ∈ Aut(X). As a special case of the previoussection, let the voltage group A be an elementary abelian group, say


A = Znp = Zp×Zp×· · ·×Zp considered as the additive group of the n-

dimensional vector space over the field GF (p). With the notation in theprevious section, we aim to construct all connected regular elementaryabelian coverings over X on which f can be lifted.

As before, let T be a spanning tree of X and let (A,A, ξ) be a locallytransitive, f -locally invariant and T -reduced Cayley voltage space onX. Then the covering graph X derived from (A,A, ξ) is connected, andhence the voltages φ(D+) generate the voltage group Zn

p . Therefore, wehave

Lemma 2.27. n ≤ β, where β is the number of cotree edges of X.

As before, we denote by ~ξ = (ξC1 , ξC2 , . . . , ξCβ) the β-dimensional

column vector of voltages on the elementary cycles. Then, we haveM~ξ = MT (f)~ξ = (ξf#C1 , ξf#C2 , . . . , ξf#Cβ

).

Lemma 2.28. Let Φ : Zβ → Zβp be the epimorphism reducing the

factors of the product modulo p. Then ~ξ⊥ = (M~ξ)⊥ if and only if

Φ(~ξ⊥) = Φ((M~ξ)⊥).

Proof. (⇒) is clear.(⇐) : Let ~a = (a1, a2, . . . , aβ) ∈ Zβ be any vector and let Φ(~a) = ~r =(r1, r2, . . . , rβ) so that ai = pqi + ri for some integers qi, 0 ≤ ri < p forall i. Since A is an elementary abelian p-group, each nontrivial elementhas order p. Hence

∑βi=1 aiξi = 0 is equivalent with

∑βi=1 riξi = 0.

Now, the sufficiency is clear. 2

In view of Lemma 2.28, we may consider the orthogonal complementof ~ξ as a subspace of the vector space Zβ

p . Identify the (column) voltage

~ξ = (ξC1 , ξC2 , . . . , ξCβ) with a (β × n) matrix [~ξ] =

− ξC1 −− ξC2 −

· · ·− ξCβ

−

β×n

with rows ξCi∈ A = Zn

p . Call [~ξ] the matrix representation of a voltage~ξ with respect to a spanning tree T . Also, one can consider the matrixrepresentation [~ξ] as a linear transformation from Zn

p to Zβp . We denote

by C([~ξ]) and R([~ξ]) the column and row spaces of the matrix [~ξ], re-

spectively. Then R([~ξ]) = 〈ξC1 , ξC2 , . . . , ξCβ〉 = A and the orthogonal

2.7. Liftings: case of elementary abelian CT(p) 71

complement ~ξ⊥ is nothing but the null space N ([~ξ]t) of the transposed

matrix [~ξ]t of [~ξ], that is,

~ξ⊥ = ~a = (a1, a2, . . . , aβ) ∈ Zβp | ~a · ~ξ = [~ξ]t~a = 0 = N ([~ξ]t).

The next lemma is obvious.

Lemma 2.29. Let M = MT (f) and [~ξ] be the matrix representations

of an automorphism f ∈ Aut(X) and a voltage ~ξ, respectively. Then the

product M [~ξ] is the matrix representation of f#~ξ = (ξf#C1 , ξf#C2 , . . . , ξf#Cβ

).Moreover, we have

(f#~ξ)⊥ = N ((M [~ξ])t) = ~a ∈ Zβ

p | (M [~ξ])t~a = 0.

-¾

¼

*

6

Znp

(M [~ξ])t

[~ξ]t

[~ξ]Zβ

p = C([~ξ])⊕N ([~ξ]t)

[f#~ξ] = M [~ξ] 6

Zβp = C(M [~ξ])⊕N ((M [~ξ])t)

M = MT (f)

Figure 2.8: Matrix representations of voltages ~ξ and f#~ξ

For a relation between the matrix representations of a voltages ~ξand f#

~ξ = (ξf#C1 , ξf#C2 , . . . , ξf#Cβ), see Figure 2.8.

Theorem 2.30. Let X be a connected graph with cycle rank β, T be aspanning tree and let (A,A, ξ) be a T -reduced locally transitive Cayleyvoltage space on X with an elementary abelian group A = Zn

p . Thenthe following statements are equivalent for an f ∈ Aut(X).

(1) f lifts to an automorphism of the derived graph Xξ,

(2) ~ξ⊥ = (M~ξ)⊥, that is, the orthogonal complements of the T -reduced

vectors ~ξ and M~ξ, where M = MT (f), are identical.


(3) N ([~ξ]t) = N ([~ξ]tM t).

(4) R([~ξ]t) = R([~ξ]tM t), or equivalently C([~ξ]) = C(M [~ξ]).

(5) The null space N ([~ξ]t) is an M t-invariant subspace of Zβp .

(6) The column space C([~ξ]) is an M-invariant subspace of Zβp .

Proof. By Theorem 2.25 (1) ⇔ (2).

(2) ⇔ (3) holds because ~ξ⊥ = N ([~ξ]t) and (M~ξ)⊥ = N ([~ξ]tM t).(3) ⇔ (4) because the row space and the null space are always

mutually orthogonal; and C(A) = R(At) for any matrix A.(3) ⇔ (5) Since f is automorphism, the matrix M t is a linear auto-

morphism on Zβp . And M t always maps N ([~ξ]tM t) to N ([~ξ]t).

(4) ⇔ (6) because the matrix M is a linear automorphism on Zβp

and it always maps C([~ξ]) to C(M [~ξ]). 2

Remark: To construct all connected elementary abelian coveringsover X on which f ∈ Aut(X) can be lifted, we employ the condition(6) in Theorem 2.30. The locally transitive, f -locally invariant andT -reduced Cayley voltage spaces (Zn

p ,Znp , ξ) on X associated with the

desired coverings can be determined using the following algorithm:

(1) Find the matrix representation M = MT (f) which is a (β × β)-matrix.

(2) Find all invariant subspaces W of M .

(3) For each M -invariant subspace W , choose a basis α = a1, a2, . . . , anwith ai ∈ Zβ

p for the subspace W .

(4) Now, define the voltages on the cotree darts by the rows of the

matrix

| | |a1 a2 · · · an

| | |

β×n

.

Recall that two covering projections pi : Xi → X, i = 1, 2, areisomorphic if there exists a graph isomorphism Φ : X1 → X2 such thatp2 Φ = p1.


Theorem 2.31. Let X be a graph and let M be a β × β matrixrepresenting an automorphism of X. Given an M -invariant subspaceW (with respect to left action), the covering graph projection derivedfrom the Cayley voltage assignment (Zn

p ,Znp , ξ) described in Remark in

page 72, where n = dim W , does not depend on the choice of the basisfor W .

If W1 and W2 are two such subspaces, then the respective coveringprojections Xi → X are isomorphic if and only if there is an automor-phism g of X such that W2 = M(g)W1.

Proof. By Theorem 2.14, there exists a group automorphism M on Znp

such that W2 = MW1. Moreover, it is proved in [MMP04, Prop. 6.3(b)]and [MP06] that M = M(g) is a representation of an automorphism ofX. 2

By Remark in page 72 and Theorem 2.31, the problem of determin-ing of all equivalence classes of f -locally invariant Cayley voltage spaces(Zr

p,Zrp, ξ) defined on X reduces to the following classical problem in

linear algebra:

Problem: Given an invertible matrix M of order β, determineall M -invariant subspaces of the β-dimensional vector space over theGalois field GF (p).

The Jordan canonical form of M = M(f) over GF (p) determinesall minimal M -invariant subspaces which are necessarily cyclic. EachM -invariant subspace can be constructed as a direct sum of the min-imal invariant ones. For a concrete matrix and a given prime p of areasonable size, the invariant subspaces can be computed using a com-puter program. However, to prove statements on all elementary abeliancoverings over a given graph one needs a more detailed understandingof the structure of common invariant subspaces with respect to a set ofmatrices. For convenience, let us recall some facts, for more details thereader is referred to [Ja53].

Invariant subspaces of one matrix: Let M be an invertible(n×n) matrix over F = GF (p) representing a linear transformation ~x 7→M~x on the vector space Fn. Denote by κM(x) = f1(x)n1f2(x)n2 · · · fk(x)nk

and mM(x) = f1(x)s1f2(x)s2 · · · fk(x)sk the characteristic and minimal


polynomials of M respectively, where fj(x) are pairwise distinct polyno-mials irreducible over F. Then the vector space Fn can be decomposedas

Fn = N (f1(M)s1)⊕N (f2(M)s2)⊕ · · · ⊕ N (fk(M)sk).

Moreover, all M -invariant subspaces can be obtained by first determin-ing the invariant subspaces of N (fj(M)sj) for each j = 1, 2, . . . , k, andthen taking their all possible direct sums. The subspace N (fj(M)sj)has dimension djnj, where dj = deg fj(x) is the degree of fj(x). Its min-imal M -invariant subspaces are cyclic of the form 〈x,Mx, . . . , Mdj−1x〉,where x ∈ N (fj(M)). Moreover, each of such minimal cyclic invariantsubspaces in N (fj(M)) defines an increasing sequence of nested invari-ant subspaces of length at most sj and at least one of such sequenceshas length sj. If nj > sj then a variety of pairwise disjoint minimalcyclic subspaces exists in N (fj(M)sj), while if nj = sj then a minimalM -invariant subspace in N (fj(M)sj) exists uniquely. In particular, ifnj = sj = 1 then N (fj(M)sj) itself is a minimal M -invariant subspace.Consequently, if κM(x) = mM(x) with nj = sj = 1, then N (fj(M)),j = 1, 2, . . . , k are the only minimal M -invariant subspaces, and all theothers are direct sums of these minimal ones.

Common invariant subspaces of more than two matrices:Let Mi ∈ GL(n,F) (i = 1, 2, . . . , `) be invertible matrices such thatG = 〈M1,M2, . . . , M`〉 is a finite subgroup of GL(n,F).

As concerns finding G-invariant subspaces, we recall the followingconsequence of a theorem by Maschke (see [AB90, page 116]).

Theorem 2.32. (Maschke) Let G ≤ GL(n,F) be a group of lin-ear transformations over a finite field F of characteristic p and letgcd(|G|, p) = 1. Then the space Fn decomposes uniquely into a directsum of minimal G-invariant subspaces and every G-invariant subspaceis a direct sum the minimal ones.

A minimal G-invariant subspace may not be a minimal invariantsubspace for neither of the generators Mi. However, for each generatorMi of G, an Mi-invariant subspace is a direct sum of the minimal Mi-invariant subspaces. In case when Maschke theorem holds, we have toidentify minimal common invariant subspaces with respect to the set ofmatrices Mi, i = 1, 2, . . . , `. The remaining case gcd(p, |G|) 6= 1 couldbe technically more difficult to analyze.


An algorithm for determining the G-invariant subspaces requiresfirst to identify the minimal polynomials of each of Mi and to factorizethem into irreducible factors. There is an exhaustive literature on thistopic, see for instance [LN84]. Since mMi

(x) divides xmi − 1, where mi

is the order of Mi, it is worth to find the irreducible factors of xmi − 1over Zp. Such method is described in [LN84, Theorem 2.45].

Example 2.17. (All elementary abelian coverings of K3,3 on which anautomorphism group G = 〈r, `〉 lifts:) In Example 2.15 we have inves-tigated locally G-invariant voltage spaces defined on K3,3 with cyclicvoltage group, where G is a group generated by the automorphismsr = (1, 2, 3, 4, 5, 6) and ` = (2, 4, 6). In other words, we have classi-fied the regular coverings ϕ of K3,3 with cyclic covering transformationgroup CT(ϕ) on which G lifts. In this example we classify the regularcovers of K3,3 with elementary abelian covering transformation groupon which G lifts.

The characteristic and the minimal polynomials for the matrix R =MT (r) are

κR(x) = mR(x) = (x− 1)(x + 1)(x2 − x + 1).

As concerns the matrix L = MT (`), its characteristic and the minimalpolynomials are

κL(x) = (x2 + x + 1)2, mL(x) = x2 + x + 1,

respectively. Since the orders of R and L are 6 and 3, and |G| =|〈R, L〉| = 18 = 2 ·32, we first assume p > 3. Then by Maschke theoremthe group G = 〈R, L〉 is completely reducible, which means that the 4-dimensional vector space V over GF (p) can be decomposed into a directsum of minimal G-invariant subspaces and every G-invariant subspaceis a sum of the minimal ones. A routine calculation shows that thequadratic polynomials (x2 − x + 1) and (x2 + x + 1) decompose intoproducts of linear factors if and only if p ≡ 1(mod 3). Moreover, ifp ≡ 1(mod 3) then

x2 − x + 1 = (x− λ)(x + λ2) and x2 + x + 1 = (x + λ)(x− λ2),

where λ3 ≡ −1(mod p).


Case of p ≡ −1(mod 3). Since there are no 1-dimensional L-invariant subspaces, V is indecomposable or V = U1 ⊕ U2 decomposesinto two minimal G-invariant subspaces of dimension 2. A good can-didate to test is an R-invariant space U1 = 〈(1, 1, 1,−2)t, (1,−1, 1, 0)t〉generated by two eigenvectors ~v1 = (1, 1, 1,−2)t and ~v2 = (1,−1, 1, 0)t

corresponding to eigenvalues ±1 of R. A direct computation shows thatLU1 = U1, hence U1 is a G-invariant subspace. The complementaryG-invariant subspace was identified as U2 = 〈(1, 0,−1, 0)t, (0, 1, 1, 0)t〉.There are no other non-trivial G-invariant subspaces.

Case of p ≡ 1(mod 3). In this case there are two 1-dimensionalL-invariant spaces W1(λ) and W2(λ) corresponding to eigenvectors~w1 = (1,−λ2,−λ, 0)t and ~w2 = (1, λ, λ2, 0)t, respectively. A directcomputation proves that the same vectors are eigenvectors for R aswell. Thus W1(λ) and W2(λ) are G-invariant and there are no other1-dimensional G invariant subspaces, since the R-invariant subspaces〈~v1〉, 〈~v2〉 corresponding to the eigenvalues ±1 are not preserved byL. On the other hand, the 2-dimensional R-invariant space U1 =〈(1, 1, 1,−2)t, (1,−1, 1, 0)t〉 = 〈~v1〉 ⊕ 〈~v2〉 is L-invariant as well (as canbe shown by a direct computation). Hence the decomposition onto min-imal G-invariant subspaces reads as follows: V = U1⊕W1(λ)⊕W2(λ).

All G-invariant subspaces together with the corresponding G-invariantvoltages are given in Table 2.1.

The cases of p = 2 and 3 are left to the reader.

Determining isomorphism classes. On the set of all the G-invariant subspaces of V, we may employ Theorem 2.31 to determine theisomorphism classes of coverings. To do this, we need to investigate anaction of Aut(K3,3) on the invariant subspaces. Since G acts regularlyon arcs, we observe that the stabilizer of the arc 12 is 〈(3, 5), (4, 6)〉.It follows that Aut(K3,3) decomposes onto left cosets as Aut(K3,3) =G ∪ (4, 6)G ∪ (3, 5)G ∪ (3, 5)(4, 6)G. Set α = (3, 5) and β = (4, 6).It is sufficient to investigate actions of the generators on the set ofinvariant subspaces. Their associated T -reduced matrices A = MT (α)


and B = MT (β) are

A =

1 −1 0 00 −1 0 00 0 −1 00 1 1 1

and B =

0 0 −1 −10 1 −1 00 0 −1 0

−1 0 1 0

,

respectively. The action on a column space U is defined by left multi-plication on a matrix [U ] whose columns form a base of U . Technicallythe question whether M [U ] = [U ] transfers to a problem of solvingn systems of linear equations, where n is the dimension of U . Sinceboth matrices MT (g) and MT (g) are regular, we may deal with thesubspaces of dimension one, two and three separately. Direct compu-tations show that AV1(λ) 6= V2(λ), BV1(λ) 6= V2(λ), AV2(λ) 6= V1(λ)and BV2(λ) 6= V1(λ). As concerns dimension 2 we have

AU1 = A

1 11 −11 1

−2 0

=

0 2−1 1−1 −1

0 0

.

Then AU1 = U2 if and only if the following two systems of linearequations have solutions:

0 2−1 1−1 −1

0 0

[xy

]=

10

−10

,

0 2−1 1−1 −1

0 0

[xy

]=

0110

.

The solutions do exist, they are x = y = 1/2 and x = −1, y = 0,respectively. This shows that the voltage assignments attached withU1 and U2 determine isomorphic coverings. Finally, we compute C =A[U1⊕V1(λ)] and form the three systems of equations C~x = ~b1, C~x = ~b2

and C~x = ~b3, where [~b1,~b2,~b3] = [U1 ⊕ V2(λ)]. Similarly as above,A takes U1 ⊕ V1(λ) → U1 ⊕ V2(λ) if and only if each of the abovethree systems of linear equations has a solution. In particular, for~b1 = [1, 1, 1,−2]t ∈ U1 we get the following system of equations

0 2 1 + λ2

−1 1 λ2

−1 −1 λ0 0 −λ2 − λ

xyz

=

111

−2

,


which has no solution for any p ≡ 1(mod 3).

We summarize the results as follows.

Let X be a connected p-elementary abelian covering over K3,3 such thatthe fibre-preserving group acts regularly on arcs. Then for p > 3, allsuch coverings over K3,3 are described in Table 2.1 by means of theassociated voltage spaces. Except the dimension two, different volt-age assignments describe nonisomorphic coverings. In particular, thereare two non-isomorphic coverings if p ≡ −1(mod 3), and six if p ≡1(mod 3).

Exercises

2.7.1. Let K3,3 be the complete bipartite graph with the bipartition V =1, 3, 5 ∪ 2, 4, 6. Determine equivalence classes of G-invariant ele-mentary abelian voltage spaces on K3,3 for the singular primes p = 2, 3,where G is generated by the automorphisms r = (1, 2, 3, 4, 5, 6) and` = (2, 4, 6).

2.7.2. Determine equivalence classes of Aut(X)-invariant elementary abelianvoltage spaces on K3,3.

2.7.3. Determine equivalence classes of Aut(X)-invariant elementary abelianvoltage spaces on X for X ∈ D(3), K4, Q3,O3.


dim Invariant [ξC1 ]t [ξC2 ]

t [ξC3 ]t [ξC4 ]

t conditionsubspace

1 V1(λ) [1] [−λ2] [−λ] [0]p ≡ 1 (3)

λ3 ≡ −1(p)

1 V2(λ) [1] [λ] [λ2] [0]p ≡ 1 (3)

λ3 ≡ −1(p)

2 U1

[11

] [1

−1

] [11

] [ −20

]none

2 U2

[10

] [01

] [ −11

] [00

]none

3 U1 ⊕ V1(λ)

111

1−1−λ2

11

−λ

−2

00

p ≡ 1 (3)

λ3 ≡ −1(p)

3 U1 ⊕ V2(λ)

111

1−1

λ

11

λ2

−2

00

p ≡ 1 (3)

λ3 ≡ −1(p)

4 V

1000

0100

0010

0001

none

Table 2.1: G-invariant voltage spaces, Case p > 3. (Here a ≡ b(n)means a = b(mod n))

Chapter 3

Applications

3.1 Enumeration of Graph Coverings

We note that the same graph X can cover the base graph X in severaldifferent ways. We are interested in enumerating covering projectionsnot the covering graphs and the number being computed always referto nonisomorphic coverings.

In what follows we shall continue the discussion started in Secion ??.Let Iso (X; r) denote the number of r-fold coverings of a graph X andlet Isoc (X; r) denote the number of connected ones. Let β be the Bettinumber β(X). Recall a T -reduced permutation voltage assignment canbe considered as a β-tuple of permutations in Sym(r) by labeling thepositive arcs in D(X) − D(T ) as e1, e2, . . . , eβ, and the set all r foldcoverings CT (X; r) can be identified as

CT (X; r) = Sym(r)× Sym(r)× · · · × Sym(r), (β factors).

The number is determined CT (X; r) is determined in Theorem 2.13.

Next, we discuss a computation of the number Isoc (X; r) of con-nected r-fold coverings of X. It is well known (see for example [Ma91])in topology that there exists a one-to-one correspondence between theconnected r-fold coverings of X and the conjugacy classes of index rsubgroups in the fundamental group of X. Indeed, Let p1 : X1 → X

81

82 Chapter 3. Applications

β r = 1 r = 2 r = 3 r = 4 r = 51 1 2 3 5 72 1 4 11 43 1613 1 8 49 681 147214 1 16 251 14491 17308615 1 32 1393 336465 2073883056 1 64 8051 7997683 24883501301

Table 3.1: The number Iso (X; r) for small r and small β

and p2 : X2 → X be two isomorphic coverings. That means that thereis a graph isomorphism Φ such that p1 = Φp2. Since Φ is fibre preserv-ing mapping consistent with the right action of the fundametal groupπ(X, x0) it follows that the images p∗1π(X1, x0) and p∗2π(X2, Φ(x0)) areconjugate in π(X, x0). Conversely, if π(X1, x0) = π(X1, y0) are conju-gate by g, where x0 and y0 belong to a fibre over x0 then the conjuga-tion by g gives rise to an isomorphism between coverings. This way theproblem of enumeration of connected graph coverings over X with theBetti number β translate to a group theoretical problem of countingconjugacy classes of subgroups of given index.

The number αF(β)(r) of index r subgroups of a free group of rankβ was determined by Hall [Ha49] as follows.

Theorem 3.1. (Hall 1949) Let F(β) be the free group generated by βelements and let αF(β)(r) be the number of index r subgroups in F(β).Then,

αF(β)(r) = r(r!)β−1 −r−1∑j=1

((r − j)!)β−1 αF(β)(j) with αF(β)(1) = 1.

Proof. We present a combinatorial proof. First, recall that Fβ =π1(Bβ, ∗) is a free group generated by β elements and all n-fold (con-nected or disconnected) coverings of a bouquet Bβ of β cycles (a onevertex graph with β loops and no semiedges) are determined by per-mutation voltages in

Sn × · · · × Sn, (β times).

3.1. Enumeration of Graph Coverings 83

(i) Given any voltage σ = (σ1, . . . , σβ), let the orbit of 1 under theaction of Gσ = 〈σ1, . . . , σβ〉 on F = 1, 2, . . . , r be

O = 1, b2, . . . , bt,and let VO be the corresponding orbit in the covering Bσ

β −→ Bβ.Let V be the n-element set of vertices and α be a labelling α :F → V taking 1 7→ v1, where v1 ∈ VO is fixed. Each σ and eachα gives a subgroup π1(B

σβ , v1) of Fβ of index t.

(ii) How many 1-similarity classes of σ are there? In other words,how many labellings α : F → V taking 1 → v1 and preserving Oset-wise does exist? There are (n − 1) · · · (n − t + 1) choices oflabels in O and there are (n− t)! choices of labels for vertices inV \VO. Note that all σ1, . . . , σβ in σ should have the same lettersfor the orbit of 1, but free for the letters which do not in the orbitof 1. Hence, in total, there are

(n− 1) · · · (n− t + 1)[(n− t)!]β = (n− 1)![(n− t)!]β−1

permutations in the 1-similar class of σ.

(iii) Let SFβ(t) denote the number of subgroups of Fβ of index t.

Since each subgroup of index t is induced by the same numberof β-tuples of permutations (in its 1-similar class) among (n!)β

permutations in (Sn)β, we have,

(n!)β =n∑

t=1

(n− 1)![(n− t)!]β−1SFβ(t).

Divide by (n− 1)! to get

n(n!)β−1 =n−1∑t=1

[(n− t)!]β−1SFβ(t) + SFβ

(n).

2

Liskovets [Li71] computed the number Isoc (X; r) of conjugacy classesof subgroups of index r of F(β) in terms of the numbers αF(β)(m) form|r.


Theorem 3.2. The number Isoc (X; r) is equal to the number of con-jugacy classes of index r subgroups in the free group generated by βelements. It is given by the formula

Isoc (X; r) =1

r

∑

m|rαF(β)(m)

∑

d| rm

µ( r

md

)d(β−1)m+1,

where µ is the number-theoretic Mobius function.

Proof. By Theorem 1.8 we have

Isoc(X; r) =1

r

∑

`| r` m=r

∑K<mG

Epi(K, Z`),

where the second sum ranges through all subgroups K of Fβ of index m.By Theorem 3.1 there are αF(β)(m) such subgroups. By Theorem 2.4these subgroups are free groups of rank (β − 1)m + 1. Finally, byCorollary 1.11 we have

Epi(K, Z`) =∑

d|`µ

(`

d

)d(β−1)m+1.

Inserting in the above formula we get the required result. 2

Later, Hofmeister [Ho98] and Kwak and Lee [KL96] independentlyfound another combinatorial enumeration formulae for the number ofconnected r-fold coverings of a graph.

Let CT (X; H) denote the set of T -reduced H-voltage assignmentsof X. Any regular r-fold covering of X is isomorphic to an derivedcovering p : X ×ξ H → X for a group H of order r and for a T -reduced Cayley voltage assignment ξ in CT (X; H). Moreover, if thegraph X ×ξ H is connected, then the group H is isomorphic to thecovering transformation group.

We now, let Iso (X; H) denote the number of regular H-coverings,and Isoc (X; H) the number of the ones that are connected. Similarly,


β r = 1 r = 2 r = 3 r = 4 r = 5 r = 61 1 1 1 1 1 12 1 3 7 26 97 6243 1 7 41 604 13753 5042434 1 15 235 14120 1712845 3715154545 1 31 1361 334576 207009649 2685307712716 1 63 7987 7987616 24875000437 193466859054994

Table 3.2: The number Isoc (X; r) for small r and small β

we let IsoR(X; r) denote the number of regular r-fold coverings regard-less of the group H involved, and of course IsocR(X; r) denotes thenumber of the ones that are connected.

It is not difficult to show that the components of any regular cover-ing X×ξ H → X are isomorphic as coverings of X, and any two isomor-phic connected regular coverings of X must have isomorphic coveringtransformation groups. Moreover, any two regular coverings of X of thesame folding number are isomorphic if and only if their components areisomorphic as coverings. Notice that each component of an H-coveringof X is an S-covering of X for some subgroup S of H. The followingtheorem of Kwak et al. [KCL98] lists some basic counting propertiesabout regular coverings.

Theorem 3.3. (1) For any r ∈ N, IsoR(X; r) =∑

d|rIsocR(X; d).

(2) For any r ∈ N, IsocR(X; r) =∑H

Isoc (X; H), where the sum is

over all nonisomorphic groups of order r.

(3) For any finite group H, Iso (X; H) =∑S

Isoc (X;S), where the

sum is over all nonisomorphic subgroups of H.

(4) For any finite groups H and K with (|H|, |K|) = 1,

Iso (X; H ⊕K) = Iso (X; H) Iso (X; K), and

Isoc (X; H ⊕K) = Isoc (X; H) Isoc (X; K),


where H ⊕K is the direct sum of two groups H and K.

To complete the enumeration of the regular coverings, we need tofind some computational formulae for Isoc (X; H). Note that the setCT (X; H) of normalized H-voltage assignments of X can be identifiedas

CT (X; H) = H ×H × · · · ×H, (β factors).

Let

X(H; r) = (g1, g2, . . . , gr) ∈ Hr | g1, g2, . . . , gr generates H.A β-tuple g = (g1, g2, . . . , gβ) yields a connected covering if and

only if g ∈ X(H; β). Under the coordinatewise Aut(H)-action on theset X(H; β), any two elements in X(H; β) belong to the same orbit ifand only if they yield isomorphic connected H-coverings, by Theorem??. Since the Aut(H)-action on X(H; β) is free, one can obtain thefollowing theorem from Burnside’s Lemma.

Theorem 3.4. For any finite group H,

Isoc (X; H) =|X(H; β)||Aut(H)| .

To complete the enumeration of IsoR(X; r) and IsocR(X; r), we needto determine |Aut(H)| and |X(H; β)|. For any finite abelian group H,Kwak et al. [KCL98] computed |Aut(H)| and |X(H; β)|.

Now, we introduce a formula to compute the number |X(H; β)| forany finite group H in terms of the Mobius function defined on thesubgroup lattice of H. The Mobius function assigns an integer µ(K)to each subgroup K of H by the recursive formula

∑H≥K

µ(H) = δK,H =

1 if K = H,

0 if K < H.

Jones [Jo95, Jo99] used such function to count the normal subgroupsof a surface group or a crystallographic group, and applied it to countcertain covering surfaces. We see that

|H|β = |CT (X; H)| =∑K≤H

|X(K; β)|.


Now, it comes from the Mobius inversion that

|X(H; β)| =∑K≤H

µ(K)|CT (X; K)| =∑K≤H

µ(K)|K|β.

The following theorem comes from Theorem 3.4.

Theorem 3.5. For any finite group H,

Isoc (X; H) =1

|Aut(H)|∑K≤H

µ(K)|K|β.

The cyclic group H = Zr has a unique subgroup Zd for each ddividing r, and has no other subgroups. The Mobius function on thesubgroup is µ(Zd) = µ(r/d) (the number-theoretic Mobius function)and |Aut(Zr)| = ϕ(r) (the Euler ϕ-function), so we have

Isoc (X;Zr) =1

ϕ(r)

∑

d|rµ

(r

d

)dβ.

Theorem 3.6. For any r = ps11 ps2

2 · · · ps`` (a prime factorization), the

number of connected Zr-coverings of X is

Isoc (X;Zr) =

0 if β = 0,

∏i=1

p(β−1)(si−1)i

pβi − 1

pi − 1if β ≥ 1,

and the total number of Zr-coverings of X is

Iso (X;Zr) =

1 if β = 0,

∏i=1

(si + 1) if β = 1,

∏i=1

(1 +

(pβi − 1)(p

si(β−1)i − 1)

(pi − 1)(pβ−1i − 1)

)if β ≥ 2 .

For the dihedral group Dr of order 2r, a similar computation givesthe following [KCL98]:


Theorem 3.7. For any r ≥ 3, the number of connected Dr-coveringsof X is

Isoc (X; Dr) =(2β − 1

) ∏i=1

p(mi−1)(β−2)i

pβ−1i − 1

pi − 1,

where pm11 · · · pm`

` is the prime decomposition of r.

Even though we have a general computational formula for Isoc (X; H),if the lattice of subgroups of H is complicated, it is not easy to eval-uate its exact value. However, if H is abelian, one can derive anotherenumeration formula for Isoc (X; H), one which does not involve thelattice structure of the subgroups of H. Our next result follows fromTheorem 3.4.

Theorem 3.8. Let t1, . . . , t` and s1, . . . , s` be natural numbers withs1 > · · · > s`. Then, the number of connected ⊕`

h=1thZpsh -coverings ofX is

Isoc (X;⊕`h=1thZpsh ) = pf(β,ti,si)

t∏i=1

pβ−i+1 − 1

∏j=1

tj∏

h=1

ptj−h+1 − 1

,

where t = t1 + · · ·+ t`, p is prime and

f(β, ti, si) = (β − t)

(∑

i=1

ti(si − 1)

)+

`−1∑

i=1

ti

∑

j=i+1

tj(si − sj − 1)

.

Now, one can compute the number Iso (X; H) for any finite abeliangroup H by using Theorems 3.3((c),(d)) and 3.8 repeatedly if necessary.For some abelian groups H and small β, the numbers Isoc (X; H) andIso (X; H) are listed in Table 3.3.

Corollary 3.9. For any ` ≥ 1, the number of connected `Zp-coveringsof X is

Isoc (X; `Zp) =(pβ − 1)(pβ−1 − 1) · · · (pβ−`+1 − 1)

(p` − 1)(p`−1 − 1) · · · (p− 1),


Isoc Isoβ (p, q) Zp3 ⊕ Zp Zq2 Zp3 ⊕ Zp ⊕ Zq2 Zp3 ⊕ Zp Zq2 Zp3 ⊕ Zp ⊕ Zq2

1 (2, 3) 0 1 0 4 3 122 (2, 5) 6 30 180 32 37 11843 (3, 5) 1404 775 1088100 2757 807 22248994 (3, 7) 126360 137200 1695792000 161451 137601 22215819051

Table 3.3: The number Isoc (X; H) and Iso (X; H) for some H andsmall β

β r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 81 1 1 1 1 1 1 1 12 1 3 4 7 6 12 8 153 1 7 13 35 31 91 57 1554 1 15 40 155 156 600 400 13955 1 31 121 651 781 3751 2801 118116 1 63 364 2667 3906 22932 19608 971557 1 127 1093 10795 19531 138811 137257 7880358 1 255 3280 43435 97656 836400 960800 63477159 1 511 9841 174251 488281 5028751 6725601 5095597110 1 1023 29524 698027 2441406 30203052 47079208 408345795

Table 3.4: The number of index r subgroups in ⊕β1Z

and the total number of `Zp-coverings of X is

Iso (X; `Zp)=1 +∑

h=1

(pβ − 1)(pβ−1 − 1) · · · (pβ−h+1 − 1)

(ph − 1)(ph−1 − 1) · · · (p− 1).

(Again, we note that Hofmeister [Ho95a] independently computed thenumber Isoc (X; `Zp) by an another method.)

We note two facts: (1) The number of connected `Zp-coveringsof a connected graph X is equal to the number of the `-dimensionalsubspaces of the β-dimensional vector space over the field GF (p) (see[To88, Ch 1.4]). (2) For a connected H-covering p : X → X, the im-age p∗(π1(X)) of the fundamental group of the covering graph X is anormal subgroup of the fundamental group π1(X) of the base graph X,and the quotient group π1(X)/p∗(π1(X)) is isomorphic to H. Now, itis not hard to show that the number

∑H

Isoc (X; H) =∑H

|X(H; β)||Aut(H)| ,


β r = 1 2 3 4 5 6 7 8 9 10 111 1 1 1 1 1 1 1 1 1 1 12 1 3 4 7 6 15 8 19 13 21 123 1 7 13 35 31 119 57 211 130 259 1334 1 15 40 155 156 795 400 1955 1210 2805 14645 1 31 121 651 781 4991 2801 16771 11011 29047 16105

Table 3.5: The number IsocR(X; r) for small r and small β

β r = 1 2 3 4 5 6 7 8 9 10 111 1 2 2 3 2 4 2 4 3 4 22 1 4 5 11 7 23 9 30 18 31 133 1 8 14 43 32 140 58 254 144 298 1344 1 16 41 171 157 851 401 2126 1251 2977 14655 1 32 122 683 782 5144 2802 17452 11133 29860 16106

Table 3.6: The number IsoR(X; r) for small r and small β

where H runs over all nonisomorphic abelian groups of order r, is equalto the number of index r subgroups in the free abelian group Z⊕· · ·⊕Zgenerated by β elements. For small r and small β, these numbers arelisted in Table 3.4.

If one can have the classification of the groups of order r, thenIsoR(X; r) and IsocR(X; r) can be computed by Theorem 3.3 (a) and(b). For example,

Iso R(X; p2) = Isoc (X; 2Zp) + Isoc (X;Zp2) + Isoc (X;Zp) + 1

=(pβ − 1)(pβ−1 − 1)

(p2 − 1)(p− 1)+ p(β−1)p

β − 1

p− 1+

pβ − 1

p− 1+ 1.

For small r and small β, these numbers are listed in Tables 3.5 and 3.6.

3.2 Some applications of graph coverings

In this section we show couples of examples to demonstrate a usefulnessof a covering construction

3.2. Some applications of graph coverings 91

3.2.1 Existence of regular graphs with large girth

A classical problem in graph theory reads as follows. Given any twointegers k and g, show that there exists a k-valent graph of girth atleast g. In what follows we employ graph coverings to construct suchgraphs.

Proposition 3.10. Given integers k ≥ 2 and g ≥ 3 there exists asimple vertex-transitive k-valent graph of girth at least g.

Proof. Denote by Tk(m) a complete rooted k-valent tree of radius m,whose pendant vertices are incident with k−1 semiedges. Take a regulark-edge-coloring of Tk(m) by colours 1, 2, . . . , k. Let τi, i = 1, 2, . . . , k,be an involution acting on vertices of Tk(m) transposing the verticesjoined by edges colored by i. Set G = 〈τ1, τ2, . . . , τk〉 and let X =Cay(G, τ1, τ2, . . . , τk) be the Cayley graph based at G determinedby the set of generators τ1, τ2, . . . , τk. Clearly, X is vertex-transitive,simple and k-valent. To estimate the girth of X, let C be a cycle oflength d. Then there is a corresponding identity w = τi1τi2 . . . τid = 1such that τij 6= τij+1

. If v is a root of Tk(m) then v · τi1τi2 . . . τimτim+1

is a pendant vertex of Tk(m), and τi1τi2 . . . τit for 1 ≤ t ≤ m takes vto a vertex at distance t from v. Since v · w = v we need to add inτi1τi2 . . . τimτim+1 a sequence made of at least m involutions τi to map vonto v. Thus the length d of C is at least 2m + 1. 2

In fact one can prove that the Cayley graph X constructed aboveregularly covers the tree Tk(m) with the covering transformation groupCT(p) ∼= StabG(v).

3.2.2 Large graphs of given degree and diameter 2

A vertex/diameter problem is to find the largest (possibly regular) graph(in the number of verticies) which has a given degree (vertex-valency)and a given diameter. For simplicity, we restrict our considerations tographs with diameter 2. For a given number d we need to find the num-ber v(d) of vertices of the largest graph having degree d and diameter2. An easy observation gives that the size of a distance spanning treecannot exceed v(d) ≤ d2 + 1.


There is a dual problem to find a minimum number of vertices µ(d)of a graph of girth 5 and valency d. Such a graph contains a completed-valent tree of depth two, thus we have v(d) ≤ d2 + 1 ≤ µ(d). Thegraphs for which v(d) = d2 + 1 = µ(d) are called Moore graphs. It iswell-known [Bi74] that Moore graphs can exist only for d = 1, 2, 3, 7, 57.Moore graphs are known for d = 1, 2, 3 and 7, namely, they are K2, thepentagon, the Petersen graph and the Hoffman-Singleton graph. Theproblem of existence of a 57-valent Moore graph is still open. Forgeneral d it is hard to find good lower bounds for v(d). The followingconstruction gives a lower bound for an infinite sequence of d.

Example 3.1. [SS05] Let q be a prime power of the form q = 4s+1 andlet GF (q) be the Galois field of order q. Let X be a two-vertex graph,whose vertices u, u′ are joined by q darts xg, g ∈ GF (q), and thereare s loops attached to both u,u′. Choose an orientation D+ such thatxg ∈ D+ for every g ∈ GF (q). Let `1, `2, . . . , `s be darts representingloops incident to u (`i 6= `−1

j ), let `′1, `′2, . . . , `

′s be darts representing

loops incident to u′. Choose an orientation D+ such that xg ∈ D+ forevery g ∈ GF (q). Let h be a primitive element of GF (q) and GF (q)+

be the additive group of the field. Let (G,G, ξ) be a Cayley voltagespace on X defined by setting G = GF (q)+ × GF (q)+, ξxg = (g, g2),ξ(`i) = (0, h2i) and ξ(`′i) = (0, h2i−1). Then the derived graph hasdegree d = q + 2s = (3q − 1)/2, and order 2|G| = 2q2 = 8

9(d + 1

2)2.

It has some more interesting properties listed in the following lemma.

Lemma 3.11. The derived graph X from the Cayley voltage space(G,G, ξ) in Example 3.1 has diameter 2.

Proof. Since X comes from a Cayley voltage space defined over a 2-vertex base graph, it is either vertex-transitive or the action of theautomorphism group has two orbits - the fibres over vertices. Hence Xhas a diameter 2 if and only if each vertex u′(x,y) or u(x,y) is reachableby a path P of distance at most two from the vertex u(0,0) and fromthe vertex u′(0,0). We show it in the first case. The other case is left tothe reader.

By definition of the derived graph, a path P : u(0,0) → u′(x,y) is of

length two if and only if for a given (x, y) 6= (g, g2) at least one of the

3.2. Some applications of graph coverings 93

equations (g, g2) ± (0, h2i−1) = (x, y) or (g, g2) ± (0, h2i) = (x, y) hasa solution (g, i), g ∈ GF (q)+ and i ∈ 1, 2, . . . , s. We have x = gand y − x2 = ±h2i or y − x2 = ±h2i−1. Since the set ±h2i|i =1, 2, . . . , s ∪ ±h2i−1|i = 1, . . . , s covers all the non-zero elements ofGF (q), there is i such that either the first, or the second equation holds.

Similarly, P : u(0,0) → u(x,y) is of length two if there is a solution(g1, g2) of the equation (g1, g

21)− (g2, g

22) = (x, y). We have x = g1 − g2

and y = x(g1 + g2). However, replacing gj by gj + i, we get x = g1− g2

and y = x(g1 + g2 + 2i). If x 6= 0 then since GF (q)+ is of odd order,we can always find i such that y = x(g1 + g2 + 2i). If x = 0 then eithery = ±h2i and u(0,y) is reachable by a path of length 1, or y = ±h2i−1

and it can be expressed as a sum of the form ±h2i ± h2j implying thatit is reachable by a path of length two. 2

As a curiosity we note that the least graph X comes from the fieldGF (5) and it is the Hoffman-Singleton graph determined as a regularcover over a two-vertex graph with voltages in Z5×Z5 (this descriptionwas discovered by Siagiova). See Fig. 3.1.

(2,4)

(1,1)

(0,0)

(3,4)

(4,1)

(0,4) (0,2)

Figure 3.1: The Hoffman-Singleton graph derived from (Z25,Z2

5, ξ)

3.2.3 Spectrum of a graph and its covering graphs

Let X = (D,V ; I, λ) be a graph without semiedges. A partition of itsvertex set V = V1 ∪ V2 ∪ · · · ∪ Vn is called an equitable partition if there


exists a square matrix M = (di,j) of order n, called a partition matrixsuch that for every i, j ∈ 1, 2, . . . , n and for every vertex x ∈ Vi thereare exactly di,j darts with initial vertex x terminating at some verticesof Vj.

Let V (X) = v1, v2 . . . , vm be the vertex set. The adjacency matrixA(X) = (aij) is the m × m matrix with aij = 1 if (vi, vj) ∈ D andaij = 0 otherwise. The characteristic polynomial of X, denoted byΦ(X; λ) = det(λI − A(X)), is the characteristic polynomial of A(X).

Lemma 3.12. Let X = (D, V ; I, λ) be a graph without semiedges butwith an equitable partition of vertices and let its partition matrix beM = (di,j). Then the characteristic polynomial of M divides the char-acteristic polynomial of X.

Proof. Let V = V1 ∪ V2 ∪ · · · ∪ Vn be an equitable partition of V witha square matrix M = (di,j). Let A(X) be the adjacency matrix of Xwith the vertex ordering following the equitable partition. Then A(X)can be considered as a block matrix with blocks bi,j, where the rowsum of bij is equal to di,j. Now, let λ be an eigenvalue of M with aneigenvector x = (x1, x2, . . . , xn). Then one can show that λ becomes aneigenvalue of A with an eigenvector x = (x1,x2, . . . ,xn), which is alsoa block column with blocks xj = (xj, xj, . . . , xj). A similarity worksalso for a generalized eigenvector. It implies that all eigenvalues of Mare those of A upto multiplicity. It completes the proof. 2

Corollary 3.13. Let p : X → X be a covering of graphs, let X have nosemiedges. Then V = ∪v∈V fibv is an equitable partition. In particular,the characteristic polynomial of X divides the characteristic polynomialof X.

Example 3.2. Let X = (D, V , I, λ) and X = (D, V ; I, λ) be twographs and let X → X be a covering projection derived from a locallytransitive permutation voltage assignment (F, G, ξ). For each γ ∈ G,let X(ξ,γ) denote the spanning subgraph of the graph X whose di-rected edge set is ξ−1(γ). Let V = v1, v2, . . . , vm. We define an(anti-lexicographical) order relation ≤ on V = V × F as follows: for(vi, s), (vj, t) ∈ V , (vi, s) ≤ (vj, t) if and only if either s < t or s = t andi ≤ j. Let |F | = n and let P (γ) denote the n× n permutation matrix

Flows 95

associated with γ-action, i.e., its (s, t)-entry P (γ)st = 1 if γ · s = t andP (γ)st = 0 otherwise. The tensor product A⊗B of the matrices A andB is considered as the matrix B having the element bst replaced by thematrix Abst. Now, one can show that the adjacency matrix A(X) of acovering graph X is

A(X) =∑γ∈Sn

A(X(φ,γ))⊗ P (γ).

Exercises

3.2.1. Using graph coverings prove that for every k ≥ 2 and for every g ≥ 3there are graphs of girth exactly g.

3.2.2. Prove that every cubic graph is a quotient of a cubic Cayley graph.3.2.3. Determine the least cubic Cayley graphs covering the Petersen graph

and the dodecahedral graph.3.2.4. Prove that each regular graph of even valency is a quotient of a Cayley

graph.3.2.5. Show that not every equitable partition of a graph comes from a graph

covering.3.2.6. Prove that an equitable partition is induced by a graph covering if

and only if the associated matrix M = (di,j) is symmetric.3.2.7. Using the description of Siagiova depicted on Fig. 2 compute the

eigenvalues of the Hoffman-Singleton graph.3.2.8. * Prove or disprove: Let X → X be a graph covering, X be without

semiedges. Then the sets of eigenvalues of X and X coincide.3.2.9. * What approximation of the degree/diameter problem one can can

get for d = 4, 5, 6 and diameter 2, using Cayley voltage spaces (G,G, ξ)defined on X with G abelian and X having at most two vertices.

3.2.10. * Giving a construction try to find a lower bound > d2/4 for thedegree/diameter= 2 problem for an infinite sequence of d, d 6= (3q −1)/2, q a prime power ≡ 1 mod 4.

3.3 Flows

Voltage assignments valued in an abelian group can be viewed as flows.Usually it is required that in all, or in most vertices the Kirchhoff law


is satisfied. Let A be an abelian group and X = (D,V ; I, λ) be a con-nected graph. An A-circulation is a Cayley voltage space (A,A, ξ) onX such that the Kirchhoff law

∑x∈Dv

ξx = 0 is satisfied for every vertexv ∈ V . Generally, for U ⊆ V denote by D(U) the set of darts with ini-tial vertices in U . Set D(U,U) = x ∈ D(U) | x 6= λ(x) and Iλ(x) ∈ U.Finally, set D(U, U) = D(U) \ D(U,U). In particular, D(V, V ) is theset of semiedges.

It follows that the edges underlying D(U, U) form an edge-cut sep-arating U . A bridge is an edge-cut of size one.

Lemma 3.14. Let X be a connected graph and let (A,A, ξ) be a cir-culation on X. Then

∑x∈D(U,U) ξx = 0 for any U ⊆ V .

In particular, the total sum of voltages on semiedges is 0 and ifx, λx is a bridge then ξx = ξλx = 0.

Proof. First note that∑

x∈D(U,U) ξx = 0 because ξx−1 = −ξx for all x.

Hence,∑

x∈D(U,U) ξx =∑

x∈D(U) ξx −∑

x∈D(U,U) ξx = 0− 0 = 0. 2

An A-circulation on X such that ξx 6= 0 for every x ∈ D is called anowhere-zero A-flow on X, or shortly an A-flow. By Lemma 3.14, allgraphs having bridges admit no circulation.

The following lemma is trivial but essential in its consequences.

Lemma 3.15. Sum of two A-circulations is an A-circulation.

As an application, we get the following lemma.

Lemma 3.16. Let X = (D,V, I, λ) be a graph, Y a maximum forestin X and let A be an abelian group. Let ξ : D(X \ Y ) → A be avoltage assignment such that for each component C of Y the total sumof voltages on the semiedges incident to C is 0. Then ξ extends toexactly one A-circulation ξ∗ : D(X) → A.

In particular, the number of A-circulations on X is |A|β(X)−γ(X),where β(X) is the Betti number of X and γ(X) is the number of com-ponents of X incident with semiedges.

Proof. We may assume that X is connected and Y is a spanning treeT . Let ξ : D(X \ Y ) → A be given. First, assume that no semiedges

3.3. Flows 97

exist. Let e = x0, x−10 be a cotree edge. Then there is a unique path

x1x2 · · · xm in T such that (x0x1x2 · · · xm) is an elementary cycle. Definean A-circulation on X by setting ξe(xi) = ξ(x0) and ξe(x

−1i ) = −ξ(x0)

for i = 1, 2, . . . , m and ξe(y) = 0 for y 6= xi, i = 0, 1, 2, . . . , m.

Now, we set ξ∗ =∑

e ξe, where the sum runs through all cotreeedges. Then ξ∗ is clearly an A-circulation.

Now, let X have s semiedges, say z1, z2, . . . , zs. We form a new graphX ′ by adding an extra vertex w and s new darts y1, . . . , ys attached tow. We extend λ by setting λ(zi) = yi. Thus X ′ is a graph withoutsemiedges and one can apply the above construction of ξ∗ by takingT ′ = T∪y1, z1 and a restriction of ξ on D(X\T ′). Since

∑si=1 ξyi

= 0,ξy1 = −∑s

i=2 ξyi= ξ∗(y1) as is required.

Hence there is at least one extension ξ∗ of ξ.

Assume there are two A-circualations ξ′ and ξ′′ that are extensionsof ξ. Then ξ′ − ξ′′ is an A-circulations taking value 0 on cotree darts.Then it is easy to see that ξ′ − ξ′′ is everywhere trivial, hence ξ′ = ξ′′.Lemma 3.14 gives γ(X) independent linear equations relating the valuesof ξ on semiedges of X. It follows that we have |A|β(X)−γ(X) choices forξ. 2

The following classical results were proved by Tutte.

Theorem 3.17. (Tutte 1954) If A and B are finite abelian groupsof the same order then X admits an A-flow if and only if X admits aB-flow.

Proof. We prove that the number of B-flows is the same as the numberof A-flows. Let F ⊆ E(X) and let β(F ) denote the Betti number ofthe subgraph of X induced on F and γ(F ) is the number of compo-nents containing semiedges. Denote µ = |A| = |B|. By Lemma 3.16µβ(F )−γ(F ) is the number of A-circulations of the subgraph induced byF . Equivalently, µβ(F )−γ(F ) is the number of A-circulations whose sup-port (the set of edges endowed by a non-trivial flow) is a subset of F .By the inclusion-exclusion principle we get for the number of nowhere-zero A-flows f(X, µ) =

∑F⊆E(X)(−1)|E(X)−F |µβ(F )−γ(F ). It follows that

the number of A-flows depends only on µ = |A|, and we are done. 2


Note that the polynomial f(X,µ) defined in the proof above is forgraphs without semiedges known as a flow polynomial of X. A Z-flowon X will be called a k-flow for some integer k if 0 < |ξx| < k for everydart x ∈ D. Tutte (1950) proved the following statement.

Theorem 3.18. (Tutte 1950) A graph X admits a k-flow if and onlyif it admits a Zk-flow.

Proof. Considering values of a nowhere-zero k-flow as elements of Zk

we obtain a nowhere-zero Zk-flow. Conversely, a nowhere zero Zk-flow can be interpreted a voltage assignment ξ : D → Z such that∑

x∈Dvξx ≡ 0 (mod k). If there are m semiedges, then as above join the

free ends of semiedges with free ends of the semistar stm thus forminga graph X ′ without semiedges end extend ξ on X ′ (there is a uniqueway to do it). Let Y be a component of X ′. Since ξ is a voltageassignment

∑v∈V (Y )

∑x∈Dv

ξx =∑

x,λ(x)∈E(ξx +ξλ(x)) = 0. Let us call

the sum δ(ξ) =∑

v∈V +(Y )

∑x∈Dv

ξx, where v runs through all vertices

with positive∑

x∈Dvξx a deficiency of the voltage assignment ξ. It

follows that if there is vertex u ∈ V (Y ) such that∑

x∈Duξx > 0 then

there is another vertex v 6= u in V (Y ) such that∑

x∈Dvξx < 0. In

such case, by connectivity, there is a path x1, . . . , xn joining u to v.Then we form a new voltage assignment ξ′ taking values ξ′xj

= ξxj− k

and ξ′x−1

j

= ξx−1j

+ k, for j = 1, 2, . . . , n. Otherwise we set ξ′ = ξ.

The resulting voltage assignment ξ′ satisfies the property∑

x∈Dvξx ≡

0 (mod k) and moreover δ(ξ′) < δ(ξ). It follows that iterating the aboveprocedure we eventually obtain a voltage assignment ζ with δ(ζ) = 0.By the definition, ζ is a k-flow on Y . Applying the procedure foreach connectivity component of X ′ we obtain a k-flow ξ on X ′. Therestriction of ξ onto darts of X gives the required k-flow on X. 2

In view of Theorems 3.17, 3.18, in order to decide whether there isan A-flow on a given graph X, it is sufficient to find the least k forwhich X admits a k-flow.

Three Tutte’s flow conjectures related to the existence of k-flows areas follows:

(1) 5-Flow Conjecture. Every bridgeless graph without semiedgeshas a 5-flow.

3.3. Flows 99

(2) 4-flow Conjecture. Every bridgeless graph without semiedgesnot containing the Petersen graph as a minor has a 4-flow.

(3) 3-flow Conjecture. Every bridgeless graph without semiedgesand without a 3-edge-cut has a 3-flow.

In 1981 Seymour proved that the flow number of a bridgeless graphis bounded by 6.

Theorem 3.19. (Seymour 1981) Every bridgeless graph without semi-edges has a 6-flow.

Proof. The proof can be found in Diestel [Di97, Chapter 6]. 2

The following observation is easy.

Lemma 3.20. Let p : X → X be a covering of graphs. If X admits anA-flow then X admits an A-flow as well.

Proof. If f : D → A is the flow-function then for each x ∈ fibx setf(x) = f(x). 2

In general an A-flow does not project along a covering projection.

Lemma 3.21. A cubic graph has a nowhere-zero 4-flow if and only ifit is 3-edge-colourable. In particular, Hamiltonian cubic graphs have4-flows.

Proof. If there is a 4-flow on X then by Theorem 3.17 there is a Z2×Z2-flow ξ on X. Let x,y and z be darts based at a vertex u. Assumingξx = ξy we get ξz = ξx + ξy = 0, a contradiction. Hence f(e) =f(x, x−1 = ξ(x) is a regular 3-edge-colouring.

Vice-versa, assume there is a regular colouring f of edges of X. Wemay assume that the three colours are the nontrivial elements of Z2×Z2.by 1, 2 and 3. For an edge e = x, x−1 we set ξx = ξx−1 = f(e). Sincefor each edge e = x, x−1 and for each vertex v we have ξx + ξx−1 = 0and

∑x∈Dv

ξx = 0, the mapping ξ is a Z2 × Z2-flow, and consequently,X admits a 4-flow. A Hamiltonian cycle in X is necessarily of evenlength, hence it can be properly coloured by using two colours. Theremaining edges form a perfect matching which can be coloured by thethird colour. Thus a cubic Hamiltonian graph is 3-edge-colourable. 2


The following well-known result is an immediate consequence ofLemma 3.14.

Lemma 3.22. (Parity lemma) Let C be an edge-cut in a 3-edge-coloured cubic graph. Let c1, c2 and c3 denote the number of edges inC coloured by colours 1, 2 and 3, respectively. Then c1 ≡ c2 ≡ c3 ≡|C|(mod 2).

The following theorem is related with the well-known folklore con-jecture establishing that every Cayley graph is Hamiltonian. Restrict-ing to cubic case we wish to prove at least a weaker result proving thatevery cubic Cayley graph is 3-edge-colourable. The problem can bereduced to cubic Cayley graphs coming from simple groups or groupswhich are close to simple groups. Here we present the following partic-ular result. Recall that a group A is solvable if it admits a descendingsubnormal serie A = N0 B N1 B N2 B N3 B · · · B Nm = 1 such thatNi/Ni+1 is abelian for each i = 0, 1, . . . , m− 1.

Theorem 3.23. A loopless cubic Cayley graph based on a solvablegroup is 3-edge-colourable.

Proof. Let X be a cubic Cayley graph determined by a group A andset of generators S. The generating set S can be of two types, eitherconsisting of three involutions, or S = r, r−1, ` with `2 = 1 andr 6= r−1. In the first case the edges (darts) of X are coloured naturallyby the three involutions giving a 3-edge-colouring. In the second casewe may assume that the order |r| is odd, otherwise we can colour theeven 2-factor determined by r by two colours and the remaining edgeby the third colour. Our strategy is to describe minimal quotients of Xnot containing loops, to colour such a quotient and lift the colouring.Let X be a minimal counterexample given by A = 〈r, `〉.

Let N C A be the minimal normal subgroup of A. If r /∈ N weform a quotient X = X/[N ] which is a Cayley graph given by A =A/N = 〈rN, `N〉. By the assumption X is not 3-edge-colourable. But|X| < |X| contradicting the minimality. Hence r ∈ N . By Aschbacher[Prop.xx] N = T ×T × · · ·×T , where T are isomorphic simple groups.Solvability of A implies T ∼= Zp is cyclic for some prime p, p 6= 2because r is of odd order. If ` ∈ N then A = N and ` = 1, since

3.3. Flows 101

N is odd. Hence A = N = 〈r〉 and X is formed by a p-cycle witha semi-edge attached to each vertex. This graph is easily seen to be3-edge-colourable, a contradiction. Hence ` /∈ N and N is a subgroupof index two. The generator ` acts by conjugation on N preservingthe simple direct factors as blocks. By minimality N = T × T ` orN = T = T `. In each case we may assume T = 〈r〉. If `r` = re thenre2

= r implying e2 = 1(mod p), giving solutions e = ±1. Then X isa p prism which is easily seen to be 3-edge-coloured, a contradiction.Assume r` /∈ 〈r〉. Let K = 〈`r`r−1〉 ≤ N . Since N is abelian andr ∈ N , K is centralised by r. However, (`r`r−1)` = r`r−1` is an inverseelement to the generator of K. Hence K is a proper normal subgroupof A and of N contradicting the minimality. 2

Exercises

3.3.1. Prove the Parity lemma (Lemma 3.22)3.3.2. Find an example of cubic hamiltonian graph covering a non-hamiltonian

one. Describe the projection of a hamiltonian cycle.3.3.3. Let p : X → X. Prove that the chromatic index χ′(X) ≤ χ′(X)

provided X has no loops, and for the chromatic number χ(X) ≤ χ(X)provided X has no loops and no semiedges.

3.3.4. ∗ Prove that cubic Cayley graphs with up to 120 vertices admit anowhere-zero 4-flow.

3.3.5. Compute the flow polynomial for K4, K3,3 and the Petersen graph.

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Index

t-transitive, 19

abelianization, 7action, 7

covering transformation group,28

faithful, 8regular, 8semi-regular, 8the fundamental group, 27the fundamental groupoid, 27transitive, 8

adjacency matrix, 94

Betti number, 40block, 9bridge, 96Burnside Lemma, 13

Cayley graph, 13characteristic polynomial, 94circulation, 96complete system of imprimitiv-

ity, 10constellation, 21core, 8, 9covering

canonical double, 63derived, 34graph, 23homological, 64

isomorphic, 32regular, 28transformation, 28

covering transformation group,28

dart, 2dart-reversing involution, 2

edge, 2equitable partition, 93

fibreabstract, 31

flow, 95flow conjecture, 98flow polynomial, 98fundamental cycle, 6fundamental group, 5fundamental groupoid, 4

Generalized Petersen graph, 16graph, 2

n-cube or Hypercube, 24n-semistar, 24automorphism group, 4Cayley, 13coset, 16covering, 23Coxeter, 20derived, 34

109

110 INDEX

dipole, 25generalized Petersen, 16, 57halved cube, 24Heawood, 20, 58Hoffman-Singleton, 93monopole, 35Moore, 92Octahedral, 14Octahedron, 15orbital, 18oriented, 3Pappus, 19Petersen, 55

half-arc-transitive, 18homology group, 7homotopic, 4

imprimitive, 10incidence function, 2isomorphic

G-spaces, 9constellations, 22covering, 39

Kirchhoff law, 96

lift, 42lifting, 38Lifting Criterion, 38link, 3locally A-invariant, 50locally f -invariant, 51loop, 3

Maschke Theorem, 74morphism, 3

Octahedral graphs., 14

Orbit-Stabilizer Theorem, 12orbital, 18orientation, 3

permutation representation, 8primitive, 10project, 43

representationspermutation, 8

semiedge, 3stabilizer, 8

unimodular, 60unimodular group, 60unique walk lifting, 25

valency, 2vertex, 2vertex/diameter problem, 91voltage, 31voltage group, 31

local, 31voltage space

Cayley, 31monodromy, 31permutation, 31regular, 31

walk, 4closed, 4reduced, 4

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