European Journal of Combinatorics 46 (2015) 45–50

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European Journal of Combinatorics

journal homepage: www.elsevier.com/locate/ejc

Vertex-transitive generalized Cayley graphswhich are not Cayley graphsAdemir Hujdurović a,b, Klavdija Kutnar a,b, Dragan Marušič a,b,c

a University of Primorska, FAMNIT, Glagoljaška 8, 6000 Koper, Sloveniab University of Primorska, IAM, Muzejski trg 2, 6000 Koper, Sloveniac University of Ljubljana, PEF, Kardeljeva pl. 16, 1000 Ljubljana, Slovenia

a r t i c l e i n f o

Article history:Received 13 July 2014Accepted 22 November 2014

a b s t r a c t

The concept of generalized Cayley graphs was introduced byMarušič et al. (1992), where it was asked if there exists a vertex-transitive generalized Cayley graph which is not a Cayley graph. Inthis paper the question is answered in the affirmative with a con-struction of two infinite families of such graphs. It is also proventhat every generalizedCayley graph admits a semiregular automor-phism.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

In this paper we consider generalized Cayley graphs, first introduced in [13].

Definition 1.1. Let G be a group, S a subset of G and α an automorphism of G such that the followingconditions are satisfied:

(i) α2= 1,

(ii) if g ∈ G, then α(g−1)g ∈ S,(iii) if f , g ∈ G and α(f −1)g ∈ S, then α(g−1)f ∈ S.

Then the generalized Cayley graph X = GC(G, S, α) on G with respect to the ordered pair (S, α)is the graph with vertex set G, with two vertices f , g ∈ V (X) being adjacent in X if and only ifα(f −1)g ∈ S.

E-mail address: klavdija.kutnar@upr.si (K. Kutnar).

http://dx.doi.org/10.1016/j.ejc.2014.11.0070195-6698/© 2014 Elsevier Ltd. All rights reserved.

46 A. Hujdurović et al. / European Journal of Combinatorics 46 (2015) 45–50

In other words, a vertex f ∈ G is adjacent to all the vertices of the form α(f )s, where s ∈ S.Note that (ii) implies that X has no loops, and (iii) implies that X is undirected. Also, in view of (i), thecondition (iii) is equivalent to α(S−1) = S. Namely, by letting f = 1 in (iii), we obtain α(S−1) = S, andconversely, if α(S−1) = S, then α(f −1)g ∈ S implies that α(g−1α(f )) = α(g−1)f ∈ S. If α = 1, thenwe say that GC(G, S, α) is a Cayley graph and write simply Cay(G, S). Therefore every Cayley graph isalso a generalized Cayley graph, but the converse is not true (see [13, Proposition 3.2]). A generalizedCayley graph GC(G, S, α) is connected if and only if S is a left generating set for the quasigroup (G, ∗),where f ∗ g = α(f )g for all f , g ∈ G (see [13, Proposition 3.5]).

In [13] the properties of generalized Cayley graphs relative to double coverings of graphs areconsidered and the following problem is posed, which suggests possible ways of constructing non-Cayley vertex-transitive graphs.

Problem 1.2. Are there vertex-transitive generalized Cayley graphs which are not Cayley graphs?

The line graph of the Petersen graph provides an affirmative answer to this question. Specifically,it is a non-Cayley vertex-transitive graph (see [14]), and is isomorphic to the generalized Cayley graphGC(Z15, S, α) on the cyclic groupZ15 with respect to the subset S = {1, 2, 4, 8} and the automorphismα ∈ Aut(Z15) acting according to the rule α(x) = 11x.

Motivated by Problem 1.2, symmetry properties of generalized Cayley graphs are consideredin this paper. In Section 3 it is shown that every generalized Cayley graph admits a semiregularautomorphism (see Theorem 3.4), shedding some new light on a classical conjecture regardingexistence of semiregular automorphisms in vertex-transitive (di)graphs. In Section 4 two infinitefamilies of vertex-transitive generalized Cayley graphs which are not Cayley graphs are constructed.All graphs in these two families are bicirculants (see Section 2 for the definition).

2. Preliminaries

Throughout this paper groups are finite and graphs are simple, finite and undirected. For a graphX let V (X), E(X), and Aut(X) be its vertex set, its edge set, and its automorphism group, respectively.If v ∈ V (X), then N(v) denotes the set of neighbors of v. For U ⊆ V (X) we denote with X[U] thesubgraph of X induced by U . A graph X is said to be vertex-transitive if Aut(X) acts transitively onV (X).

For a group G and α ∈ Aut(G), the set Fix(α) is defined as Fix(α) = {g ∈ G | α(g) = g}. For g ∈ G,the permutation of G induced by left multiplication by g is denoted by gL, and for any H ≤ GwewriteHL for {hL | h ∈ H}.

For a permutation group G on a set V , we say that a partition B of V is G-invariant (alternatively,an imprimitivity block system for G) if the elements of G permute the parts, that is, blocks of B, setwise.

A non-identity permutation is semiregular, in particular (m, n)-semiregular if it has m cycles ofequal length n in its cycle decomposition. An n-bicirculant (bicirculant, in short) is a graph admitting a(2, n)-semiregular automorphism. Let X be a connected n-bicirculant admitting a (2, n)-semiregularautomorphism ρ. Then ρ enables us to label the vertex and edge sets of X in the following way: V (X)can be partitioned into two subsets U = {ui | i ∈ Zn} and V = {vi | i ∈ Zn} where

ρ(ui) = ui+1 and ρ(vi) = vi+1, i ∈ Zn,

and E(X) can be partitioned into three subsets

L =

i∈Zn

{{ui, ui+l} | l ∈ L},

M =

i∈Zn

{{ui, vi+m} | m ∈ M},

R =

i∈Zn

{{vi, vi+r} | r ∈ R},

where L,M, R are subsets of Zn such that L = −L, R = −R, M = ∅ and 0 ∈ L ∪ R. We shall denote Xby BCn[L,M, R] (as in [9]). The vertices ui, i ∈ Zn, will be referred to as left vertices and the vertices

A. Hujdurović et al. / European Journal of Combinatorics 46 (2015) 45–50 47

vi, i ∈ Zn, will be referred to as right vertices. The edgeswith both endvertices being left (right) verticesare called left (right) edges, and the remaining edges are calledmiddle edges or spokes. Observe that

BCn[L,M, R] ∼= BCn[L,M + µ, R] (∀µ ∈ Zn), (⋆)

and thus we can always assume that 0 ∈ M .

3. Semiregular automorphisms in generalized Cayley graphs

It is known that every transitive permutation group contains a fixed-point-free element of primepower order (see [5, Theorem 1]), but not necessarily a fixed-point-free element of prime order (and,hence, a semiregular element); see, for instance, [3–5,7,8,10]. In 1981, Marušič asked if every vertex-transitive digraph admits a semiregular automorphism (see [12, Problem 2.4]). The existence of suchautomorphisms is important in many proofs related to some important open problems in algebraicgraph theory. For example, the hamiltonicity problem for connected vertex-transitive graphs relies onthe existence of a semiregular automorphisms (see [1,6,11]). Although not every generalized Cayleygraph is vertex-transitive, Theorem 3.4, in some sense, gives a new partial affirmative answer to thisquestion.

Throughout this section let G be a non-trivial group and α ∈ Aut(G) an automorphism of G. We letωα:G → G be the mapping defined by ωα(x) = α(x)x−1 and let ωα(G) = {ωα(g) | g ∈ G}. Noticethat Definition 1.1(ii) is equivalent to ωα(G) ∩ S = ∅.

Lemma 3.1. We have |ωα(G)| =|G|

|Fix(α)|.

Proof. Let x, y ∈ G, and suppose that ωα(x) = ωα(y), that is, α(x)x−1= α(y)y−1. This is equivalent

to α(y−1x) = y−1x, that is, y−1x ∈ Fix(α). It follows that ωα(x) = ωα(y) is equivalent to x ∈ yFix(α).This implies that x and y have the same image under ωα if and only if x and y belong to the same leftcoset of Fix(α). Hence the number of different images of ωα equals the number of different left cosetsof Fix(α) in G, that is |ωα(G)| = |G|/|Fix(α)|. �

Corollary 3.2. Let X = GC(G, S, α). Then

|S| ≤ |G| −|G|

|Fix(α)|.

Proof. Since ωα(G) ∩ S = ∅ we have S ⊆ G \ ωα(G) and the result follows from Lemma 3.1. �

Lemma 3.3. Let X = GC(G, S, α). Then Fix(α)L ≤ Aut(X).

Proof. Let g ∈ Fix(α). Since α(g) = g we have

α(gx)−1gy = α(gx)−1α(g)y = α((gx)−1g)y = α(x−1)y for all x, y ∈ G,

and hence gx is adjacent to gy if and only if x is adjacent to y, implying that gL ∈ Aut(X). �

Note that, since every non-identity permutation induced by left multiplication is semiregular, eachnon-identity element of Fix(α)L is a semiregular automorphism of X .

Theorem 3.4. GC(G, S, α) admits a semiregular automorphism which lies in GL.

Proof. If Fix(α) = {1}, then the result follows from Lemma 3.3. Suppose now that Fix(α) = {1}. Thenby Lemma 3.1, ωα(G) = G. On the other hand, by Definition 1.1(ii), ωα(G) ∩ S = ∅, implying thatS = ∅ and, consequently, X is edgeless and thus GL ≤ Aut(X). �

As alreadymentioned, not every generalized Cayley graph is necessarily a Cayley graph. As a directconsequence of Theorem 3.4, however, non-Cayley generalized Cayley graphs of prime order do notexist.

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Corollary 3.5. Every generalized Cayley graph of prime order is a Cayley graph.

In fact, Corollary 3.5 also follows directly from Definition 1.1. Namely, if G is of prime order, it iscyclic and so is Aut(G). In particular, the only automorphism of G of order 2 is inversion, and thusDefinition 1.1(ii) implies that no element of S is a square, but every element of a group of odd order isa square, and thus S is empty.

For a Cayley graph X = Cay(G, S), we denote the set of all automorphisms of G that fixes the setS with Aut(G, S), that is, Aut(G, S) = {ϕ ∈ Aut(G) | ϕ(S) = S}. It is well-known that Aut(G, S) is asubgroup of Aut(X) contained in the stabilizer of 1 ∈ V (X) (see [15]). In the case of generalized Cayleygraphs, we have a similar result. Before stating the result, let us denote with Aut(G, S, α) the set of allautomorphisms of the group G that fixes the set S and commutes with α, that is

Aut(G, S, α) = {ϕ ∈ Aut(G) | ϕ(S) = S, αϕ = ϕα}.

Theorem 3.6. Let X = GC(G, S, α). Then Aut(G, S, α) is a subgroup of Aut(X) which is contained in thestabilizer of 1 ∈ V (X).

Proof. Let ϕ ∈ Aut(G, S, α). Let x and y be two adjacent vertices of X . Then, by Definition 1.1, thereexists s ∈ S such that y = α(x)s. Since ϕ is an automorphism of G we have ϕ(y) = ϕ(α(x)s) =

(ϕα)(x)ϕ(s). Since ϕ ∈ Aut(G, S, α) it now follows that ϕ(y) = α(ϕ(x))s′ for some s′ ∈ S. Therefore,ϕ(x) and ϕ(y) are adjacent in X , and since ϕ is a bijection it follows that ϕ ∈ Aut(X). It is clear that ϕfixes the vertex 1 of X , completing the proof. �

4. Generalized Cayley bicirculants

By Theorem 3.4 every generalized Cayley graph admits a semiregular automorphism. It is ofparticular interest to study generalized Cayley graphs admitting a (2, n)-semiregular automorphism,that is, generalized Cayley bicirculants. If a group G has an automorphism α of order 2 such that Fix(α)is cyclic and has index 2 in G, then, by Lemma 3.3, every generalized Cayley graph on G with respectto α is bicirculant. In particular, this holds for G = Z4a and α defined by α(x) = (2a+ 1)x (notice that2 ∈ Fix(α) and 2 is of order 2a in Z4a). In particular, one can easily see that the following propositionholds.

Proposition 4.1. For a natural number a, let X = GC(Z4a, S, α) where α(x) = (2a + 1)x. Then X isisomorphic to the bicirculant BC2a[L,M, R], where L = {s/2 | s ∈ S, s even}, M = {(s − 1)/2 | s ∈

S, s odd}, and R = a + L.

Remark 4.2. The bicirculant BC2a[L,M, a + L] is isomorphic to GC(Z4a, S, α), where S = {2l | l ∈

L} ∪ {2m + 1 | m ∈ M} and α(x) = (2a + 1)x.

The next theorem gives an infinite family of vertex-transitive generalized Cayley graphs, which arenot Cayley graphs. It is left to the reader to check that α and S given in the statement of this theoremindeed satisfy the conditions of Definition 1.1.

Theorem 4.3. For a natural number k ≥ 1 let G = Z4(2k2+2k+1) and let X = GC(G, S, α), whereS = {±2, ±4k2, 2k2 + 2k + 1} and α(x) = (4k2 + 4k + 3)x. Then X is a non-Cayley vertex-transitivegraph.

Proof. By Proposition 4.1 and (⋆) (taking µ = −k2 − k) we have

X ∼= BC2(2k2+2k+1)[{±1, ±2k2}, {0}, {±(2k + 1), ±(2k2 + 2k)}].

Let m = 2(2k2 + 2k + 1), and let ρ = (u0 u1 . . . um−1)(v0 v1 . . . vm−1) be the (2,m)-semiregularautomorphism of X giving the bicirculant structure described above. Let U = {ui | i ∈ Zm} andV = {vi | i ∈ Zm} be the two orbits of ⟨ρ⟩. Observe that the left and right edges all lie on some 4-cycle

A. Hujdurović et al. / European Journal of Combinatorics 46 (2015) 45–50 49

whereas no spoke lies on a 4-cycle. Since the subgraphs of X induced by U and V are connected, weconclude that {U, V } is an imprimitivity block system for Aut(X).

The kernel of the action of Aut(X) induced on {U, V } is denoted by N . Then N is normal in Aut(X)and H = ⟨ρ⟩ is contained in N . Let X1 = X[U]. Then X1 ∼= Cay(Zm, {±1, ±2k2}). Since |M| = 1,it is not difficult to see that each element of N corresponds to an element of Aut(X1) (that is, N actsfaithfully onV (X1)). It follows thatN is isomorphic to a subgroup of Aut(X1). Since, by [2, Theorem1.1],X1 is a normal circulant, we have Aut(X1) ∼= Zm o Aut(Zm, {±1, ±2k2}) ∼= D2m, the dihedral group oforder 2m, showing that H ≤ N . D2m. Since |H| = m one can see that H is characteristic in N , andthus the normality of N in Aut(X) implies that H is normal in Aut(X).

Let τ :G → G be defined by the rule

τ(uj) = v(2k+1)j and τ(vj) = u(2k+1)j.

Then one can easily see that τ is an automorphism of X of order 4. Let G = ⟨ρ, τ ⟩ ≤ Aut(X).Observe that τ interchanges the two orbits of H on V (X) and hence G acts transitively on V (X). SinceH ∩ ⟨τ ⟩ = {1}, and H is normal in Aut(X) (and consequently in G), it follows that G = H o ⟨τ ⟩ andtherefore |G| = 4m. We conclude that |Aut(X)| = 2|N| ≤ 2|Aut(X1)| = 4m and thus G = Aut(X).

Suppose now that there exists a regular subgroup K ≤ Aut(X). Since X is of order 2m it followsthat |Aut(X): K | = 2, implying that 1 = τ 2

∈ K . However, τ 2 fixes the vertex u0 and thus K is notsemiregular, which is a contradiction. This proves that X is not a Cayley graph. �

In the next theorem a second infinite family of vertex-transitive generalized Cayley graphs, whichare not Cayley graphs, is given. The family consists of bicirculants of valency 6. As before it is left tothe reader to check that α and S given in the statement of the theorem indeed satisfy the conditionsof Definition 1.1.

Theorem 4.4. For an odd natural number t such that t ≡ 0(mod 5) let X = GC(Z20t , S, α), whereS = {±2t, ±4t, 5, 10t − 5} and α(x) = (10t + 1)x. Then X is a non-Cayley vertex-transitive graph.

Proof. By Proposition 4.1 and (⋆) (taking µ = −2) we have

X ∼= BC10t [{±t, ±2t}, {0, 5t − 5}, {±3t, ±4t}].

Let ρ = (u0 u1 . . . u10t−1)(v0 v1 . . . v10t−1) be a semiregular automorphism of X which generates thedescribed bicirculant structure. Let U = {u0, u1, . . . , u10t−1} and V = {v0, v1, . . . , v10t−1} be the twoorbits of ⟨ρ⟩. Observe that the left and right edges all lie on some triangle whereas no spoke lies on atriangle. Namely, if a spoke edge in X belongs to a triangle, then 5t −5 ∈ {±t, ±2t, ±3t, ±4t}, whichis impossible since by assumption t ≡ 0(mod 5). Therefore, no automorphism of X maps a left edgeor a right edge into a spoke. This implies that {U, V } is an imprimitivity block system for Aut(X).

For i ∈ {0, 1, . . . , t − 1} define the sets Ui and Vi with Ui = {uj | j ≡ i(mod t)} and Vi =

{vj | j ≡ i(mod t)}, respectively. Observe that Ui and Vi are the vertex sets of the connectedcomponents of X[U] and X[V ], respectively. Since U and V are blocks of imprimitivity for Aut(X),B = {Ui, Vi | i ∈ {0, 1, 2, . . . , t − 1}} is also an imprimitivity block system for Aut(X).

Consider the cycles in X induced by the spokes. Observe that there are ten such cycles of length 2t .Denote the vertex sets of these cycles by Ci, for i ∈ {0, 1, . . . , 9} so that ui ∈ Ci. Since automorphismsof X map spokes to spokes, it is easy to see that C = {Ci | i ∈ {0, 1, . . . , 9}} is an imprimitivity blocksystem for Aut(X).

Suppose that ϕ is a semiregular automorphism of X that maps u0 into v0. Since B and C areimprimitivity block systems for Aut(X), it follows that ϕ(C0) = C0 and ϕ(U0) = V0. Furthermore,ϕ induces an isomorphism between X[U0] ∼= Cay(Z10, {±1, ±2}) and X[V0] ∼= Cay(Z10, {±3, ±4}).There are only two isomorphisms between Cay(Z10, {±1, ±2}) and Cay(Z10, {±3, ±4}) that map0 ∈ V (Cay(Z10, {±1, ±2})) into 0 ∈ V (Cay(Z10, {±3, ±4})), and they are given as multiplicationwith ±3 (calculating modulo 10). Let λ ∈ {3, −3} be such that the isomorphism between X[U0] andX[V0] induced by ϕ acts as multiplication with λ. Then the induced action of ϕ on C has the followingcyclic decomposition ϕC

= (C0)(C5)(C1 Cλ Cλ2 Cλ3)(C2 C2λ C2λ2 C2λ3), implying that the order of ϕ isdivisible by 4.

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Since ϕ(C0) = C0, and C0 is a block of imprimitivity for Aut(X), the automorphism ϕ induces anautomorphism ϕC0 of the cycle induced by C0. However, since C0 is a cycle of length 2t and t is odd,there is no automorphism of C0 of order divisible by 4. Hence there is no semiregular automorphismmapping u0 into v0, and we can conclude that X is not a Cayley graph.

To complete the proof we need to show that X is vertex-transitive. Let r = λt + 1 where λ is apositive integer chosen in such a way that r ≡ ±3(mod 10). Define the mapping τ in the followingway:

τ(ui) = vri and τ(vi) = uri.

It is not difficult to verify that τ is an automorphism of X and that it interchanges the two orbits of⟨ρ⟩, thus X is vertex-transitive. �

5. Conclusions

We have solved the open problem proposed in [13] regarding the existence of vertex-transitivenon-Cayley generalized Cayley graphs. It transpires that generalized Cayley graphs, specifically thoseassociated with cyclic groups, are a rich and new source of non-Cayley vertex-transitive graphs.Therefore, we would like to propose a future line of research by asking for the classification of allgeneralized Cayley graphs arising from cyclic groups.

Acknowledgments

We thank the anonymous referees for carefully reading the manuscript and several helpfulsuggestions.

The first author was supported in part by ARRS, P1-0285 and proj. mladi raziskovalci. The secondauthor was supported in part by ARRS, N1-0011, P1-0285, J1-2055, Z1-4006, J1-6720 and J1-674, andin part by WoodWisdom-Net+, W 3B. The third author was supported in part by ARRS, N1-0011, P1-0285, J1-2055, J1-4021, J1-5433 and J1-6720.

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