four constructions of highly symmetric tetravalent graphs

Four Constructionsof Highly SymmetricTetravalent Graphs

Aaron Hill1 and Steve Wilson2

1DEPARTMENT OF MATHEMATICS,UNIVERSITY OF NORTH TEXAS, 1155 UNION CIRCLE #311430

DENTON, TX 76203, USAE-mail: [email protected]

2DEPARTMENT OF MATHEMATICS AND STATISTICS,NORTHERN ARIZONA UNIVERSITY, BOX 5717

FLAGSTAFF, ARIZONA 86011E-mail: [email protected]

Received April 19, 2008; Revised July 26, 2009

Published online in Wiley Online Library (wileyonlinelibrary.com).DOI 10.1002/jgt.20520

Abstract: Given a connected, dart-transitive, cubic graph, constructionsof its Hexagonal Capping and its Dart Graph are considered. In each case,the result is a tetravalent graph which inherits symmetry from the originalgraph and is a covering of the line graph.Similar constructions are thenapplied to a map (a cellular embedding of a graph in a surface) givingtetravalent coverings of the medial graph. For each construction, conditionson the graph or the map to make the constructed graph dart-transitive,semisymmetric or 1

2 -transitive are considered. C© 2012 Wiley Periodicals, Inc. J. Graph

Theory 00: 1–16, 2012

Keywords: Graph, Map, Symmetry, Capping, Dart Graph, Cubic Graph, Corners

Contract grant sponsor: National Science Foundation; Contract grant number:DMS-0139523 (to A. H.).

Journal of Graph TheoryC© 2012 Wiley Periodicals, Inc.

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2 JOURNAL OF GRAPH THEORY

1. PRELIMINARIES

A. Graphs

A graph � is a finite collection of distinct vertices V (�) together with a collection ofdistinct edges E(�), where each edge is an unordered pair of distinct vertices. A directededge, or dart, also called an arc, is an ordered pair (u, v) such that {u, v} ∈ E(�). Thecollection of darts is D(�). A graph is called bipartite provided that there is a two-coloring of the vertices so that every edge contains one vertex of each color. A graphis called n-regular if each vertex is an element of exactly n edges. A 3-regular graph isusually called cubic and a 4-regular graph is tetravalent.

A symmetry, or automorphism, of � is a permutation σ ofV (�) such that for all {u, v} ∈E(�), {uσ, vσ } ∈ E(�). The collection of symmetries of � is denoted by Aut(�), andforms a group under composition. If Aut(�) acts transitively on V (�), E(�), or D(�),then � is said to be vertex-transitive, edge-transitive, or dart-transitive, respectively. If� is bipartite and connected, then each symmetry of � is either color-preserving orcolor-reversing. Let Aut+(�) be the group of color-preserving symmetries.

Suppose that � is connected, regular, and edge-transitive. Let (u, v) ∈ D(�) and let� be the orbit of (u, v) under Aut(�). There are three possibilities.

1. � = D(�). This holds if and only if � is dart-transitive. In this case, we also say� is symmetric.

2. For all w ∈ V (�), (w, u) /∈ �. That is, � contains no darts with terminal vertex u.One can easily check that since � is edge-transitive, each vertex appears either asa terminal vertex in � or an initial vertex in �, but not both. Hence, � is bipartiteand � has no color-reversing symmetries. Moreover, Aut(�) is transitive on eachpartite set. In this case, we say � is semisymmetric.More loosely, if Aut+(�) is transitive on edges, we say � is bi-transitive.

3. Neither of the above hold. That is, � contains some dart with terminal vertex ubut does not contain every dart of �. From the edge-transitivity of �, it followsthat for each vertex z ∈ V (�), exactly half of the darts with terminal vertex z arein � and exactly half of the darts with initial vertex z are in �. Moreover, � mustbe vertex-transitive and be of even degree. In this case, we say � is 1

2 -transitive.More loosely, if � is an orientation for �, i.e., a selection of one dart from eachedge of �, such that Aut(�) is transitive on vertices and on edges, then we call �

semitransitive and we call � a semitransitive orientation of �.

An n-arc of � is directed path of length n in � in which no two of any three consecutivevertices are the same. � is said to be n-arc-transitive if Aut(�) acts transitively on n-arcs.One-arc-transitivity is equivalent to dart-transitivity and 2-arc-transitivity is a strongercondition. If a graph is 1-arc-transitive but not 2-arc-transitive, we call it dart-regular.

To see some consequences of these definitions, consider a vertex b of a dart-transitivecubic graph and its neighbors a, c, d. Because every dart is in the orbit of (a, b), every2-arc is in the orbit of (a, b, c) or (a, b, d) or both. If (a, b, c) and (a, b, d) are in the sameorbit of 2-arcs, then every 2-arc is in this orbit and so the graph is 2-arc-transitive. If not,i.e., if the graph is dart-regular, then the stabilizer H of the dart (a, b) must fix c and d, aswell. Then H stabilizes the dart (b, c) an so must fix all neighbors of c, and similarly, allneighbors of d and their neighbors and so on. Because the graph is connected, H mustfix all vertices and so be trivial.

Journal of Graph Theory DOI 10.1002/jgt

FOUR CONSTRUCTIONS OF HIGHLY SYMMETRIC TETRAVALENT GRAPHS 3

We summarize this discussion as a proposition:

Proposition 1. Let � be a cubic dart-transitive graph. Then either � is 2-arc-transitiveor the stabilizer of a dart is trivial.

The study of edge-transitive cubic graphs is well-established. Many papers dealingwith the topic have been published in the last 50 years. A census of the dart-transitivecubic graphs with up to 768 vertices has been assembled [1], as has a census of thesemisymmetric cubic graphs in the same range [3].

For tetravalant (also called quartic) graphs, not as much is known in general. A gooddeal has been done with respect to the 1

2 -transitive case, as 4 is the least possible valencefor such a graph. Papers such as [2, 4, 5, 7, 10, 11, 12] have considered and in somecases classified such graphs having specified properties; of special note here is the recentclassification of “tightly attached” tetravalent 1

2 -transitive graphs [6, 7, 14, 17]. A censusis being assembled at [13] of edge-transitive tetravalent graphs having up to 150 vertices.The constructions of this article contribute to this topic in general and this census inparticular.

B. Maps

A map M is an embedding of a graph into a surface so that each face, i.e., each connectedcomponent of the complement of the graph in the surface, is homeomorphic to a disk. Anautomorphism or symmetry of a map is a permutation of its parts (edges, vertices, faces)which preserves incidence. Aut(M) is the group of symmetries of M under composition.We say that M is rotary provided that for some face f and for some vertex v of thatface, Aut(M) contains symmetries R and S which act as one-step rotations about fand v, respectively. If M is rotary, we say it is reflexible provided that Aut(M) alsocontains symmetries which act locally as reflections. We refer the reader to [15] for moreinformation about maps and their symmetries. A map which is rotary but not reflexibleis called chiral.

C. Corners

Maps and cubic graphs have in common the idea of a corner. In a cubic graph �, a corneris any pair {a, b}, {b, c} of edges with a common vertex. In a map M, a corner is a pair{a, b}, {b, c} of edges which are adjacent in either of the two circular orders around someface. Thus, in each of these two venues, every edge belongs to exactly four corners.

We use this idea to define two related constructions, the line graph L(�) of a cu-bic graph �, and the medial graph MG(M) of a map M. In each case, the ver-tices of the new graph are the edges of � or M, and the edges of the new graphare the pairs which form corners. Then L(�) and MG(M) are both tetravalent. Thesymmetries of the cubic graph or map act also as symmetries of the correspondingconstructed graph. If � is 2-arc-transitive then L(�) is dart-transitive. If � is dart-regular, then the line graph might be dart-transitive or 1

2 -transitive; examples of bothcases exist. Similarly, if M is reflexible, then MG(M) is dart-transitive, while if Mis chiral, then MG(M) might be dart-transitive or 1

2 -transitive; examples of both casesexist.



The constructions in this article will be graphs which are fourfold and twofold coversof both L(�) and MG(M). In each case, we want to find conditions on � or M whichmake the constructed graph dart-transitive, semisymmetric, or 1

2 -transitive.The next two propositions show that in each venue, the symmetry group is transitive

on corners.

Proposition 2. If � is a dart-transitive cubic graph, then Aut(�) is transitive on itscorners.

Proof. Consider the corners {{a, b}, {b, c}} and {{x, y}, {y, z}}. Let d be the otherneighbor of b and let w be the other neighbor of y. Since � is dart-transitive, we can findσ ∈ Aut(�) so that bσ = y and dσ = w. Then because the graph is cubic, σ must senda, c to x, z in some order, and so must send the first corner to the second. �

Proposition 3. If M is a rotary map, then Aut(M) is transitive on its corners.

Proof. Consider one corner of the face f , and letC be its orbit under Aut(M). Becauseof R, we know that C contains all corners of f . Because of S, we know that C containsall corners of all faces around v. Continuing in this way, and using the connectedness ofM, we see that C contains all corners of M. �

2. CONSTRUCTIONS FOR CUBIC GRAPHS

D. Hexagonal Capping of A Cubic Graph

I. The Construction. For the rest of this section, let � be a connected, dart-transitive,cubic graph. We form a new graph �, called the Hexagonal Capping of �, or HC(�).

Define the symbols to be ordered pairs (a, i) where a ∈ V (�) and i {0, 1} though wewill write ai instead of (a, i).

The vertex set V (�) consists of all unordered pairs of symbols {ai, b j}, where {a, b} isan edge in �. Thus, � has four vertices for each edge in �. Since � is 3-regular, � hassix times as many vertices as �.

Now we describe the edges of �, which are unordered pairs of its vertices. The edgeset E(�) consists of all {{ai, b j}, {b j, c1−i}}, where {a, b} and {b, c} form a corner in �

and i, j ∈ {0, 1} (see Fig. 1).Because each edge in � is a part of four corners, � is tetravalent. It follows that � has

eight times as many edges as �.An important feature of this construction (and the origin of its name) is the presence

of certain 6-cycles in � that come from vertices in �. Let {a, b} be an edge in � withneighboring edges {a, e}, {a, f }, {b, c}, and {b, d}. There are 20 vertices in � that comefrom these five edges. These 20 vertices together with the edges that connect them in �

can be drawn as shown in Figure 2.In Figure 2, we can see four 6-cycles, each having the property that all of the vertex

labels in the 6-cycle have a common element. We will reserve the word hex for this kindof 6-cycle, and denote one hex by putting square brackets around the common element.The hexes in Figure 2 are [b1], [a0], [b0], and [a1], in order from left to right.



FIGURE 1. The edges of HC(�) at one corner of �.

FIGURE 2. A chain of four hexes.

We will refer to this configuration of four hexes as a chain of hexes. In a chain, each hexis joined to its neighbors at opposite vertices in the hex, and these four linking verticesare in fact the four vertices covering a single edge of �.

Figure 1 shows that HC(�) is a fourfold cover of the line graph of �. One can see fromFigure 2 that all vertices coming from the same edge of � are in the same component.Thus, � is connected.

Let B be the set of vertices in � whose elements have the same subscript. Let W be theset of vertices of � whose elements have different subscripts. It is clear that � is bipartitewith partite sets B and W . We call the elements of B black vertices and the elements ofW white vertices. We distinguish between two types of black vertices. The black verticeswhose elements both have subscript 0 are said to be of type 0. The black vertices whoseelements both have the subscript 1 are said to be of type 1.

II. Results. In this section, we will prove the following theorem.

Theorem 1. Let � be a dart-transitive, connected, cubic graph and � its hexagonalcapping.

1. If � is 2-arc transitive and bipartite, then � is symmetric.2. If � is 2-arc transitive but not bipartite, then � is semisymmetric.3. If � is dart-regular and bipartite, then � is 1

2 -transitive.4. If � is dart-regular but not bipartite, then � is not edge-transitive.



If we apply the construction to the 298 dart-transitive cubic graphs in Foster’s Census ascompleted in [1], we get 128 graphs that are symmetric, 35 graphs that are semisymmetric,133 graphs that are 1

2 -transitive, and only 2 graphs that are not edge-transitive.The outline of the proof is this:

1. We introduce and deal with the five “exceptional” cases in which � has smallgirth.

2. We introduce symmetries β and σ ∈ Aut(�) inherited from the construction andfrom Aut(�), respectively.In the nonexceptional cases, then, we show that:

3. The only 6-cycles in � are the hexes.4. Every symmetry φ of � acts as a symmetry φ of �.5. Aut(�) has a color-reversing symmetry δ if and only if � is bipartite.6. Aut+(�) is transitive on edges if and only if λ is 2-arc-transitive.7. We show how these propositions prove the theorem.

It is well known and not hard to check that the only dart-transitive, connected, cubicgraphs with girth less than six are these five: K4, K3,3, the cube Q3, the dodecahedron,and the Petersen graph; these have girths 3, 4, 4, 5, 5, respectively. Each of these is2-arc-transitive and, among these, only K3,3 and the cube are bipartite. We consider theseone at a time:

1. If � = K4, then � = HC(�) has the property that each black vertex has exactlythe same neighbors as one other black vertex. For instance, if the vertices of K4 area, b, c, d then the neighbors of {a0, b0} are {a0, c1}, {a0, d1}, {b0, c1}, and {b0, d1}.The vertex {c1, d1} has exactly the same neighbors.A graph having the property that some two vertices have the same set of neighborsis called unworthy. If we identify two vertices that have the same neighbors, theresulting graph is made from the octahedron by inserting one vertex of degree 2 ineach edge. The site [13] calls this construction the subdivided double, and everygraph constructed in this way is semisymmetric. In particular, this graph is listedin [13] as the subdivided double of the octahedron. It has tag C4[24, 7].

2. HC(K3,3) happens to be the skeleton of the map {4, 4}6,0. The graph is dart-transitive, as all {4, 4}b,c graphs are. It is C4[36, 4] in [13].The remaining three graphs have no simpler construction than that in this article.

3. HC(Q3) is a dart transitive graph on 48 vertices; its symmetry group is of size768. It is C4[48, 10] in [13].

4. When � is the Petersen graph, then � is a semisymmetric graph on 60 vertices;its symmetry group has size 240. It is C4[60, 10] in [13].

5. When � is the dodecahedron, then � is a semisymmetric graph on 120 vertices;its symmetry group has size 240. It is C4[120, 31] in [13].

The graph � inherits certain symmetries from the graph � and the method of construc-tion. Define β: V (�) −→ V (�) by {ai, b j}β = {a1−i, b1− j}. It is clear from the definitionof β and the construction of � that β is a symmetry of �.

Every symmetry σ of � naturally induces a symmetry σ of � defined by {ai, b j}σ ={(aσ )i, (bσ ) j}. It is easy to check that this permutation of the vertices of � is indeed asymmetry. Both of these inherited symmetries preserve colors of vertices in � and so arein Aut+(�).



It is clear that F defined by F(σ ) = σ is an injective homomorphism mapping Aut(�)

one-to-one into Aut+(�). Because Aut(�) is transitive on edges, we see that Aut+(�) istransitive on white vertices and on black vertices.

Having dealt with the five exceptional cases, we assume for the rest of the proof thatthe girth of � is at least six. Under that assumption, we have the following proposition:

Proposition 4. Every 6-cycle in � is a hex.

Proof. Every 6-cycle in � must contain three black vertices and three white vertices.Recall that each black vertex is of the form {a0, b0} (type 0) or of the form {a1, b1} (type1).

Suppose � has a nonhex 6-cycle that has three black vertices of type i. It is easy to checkthat this 6-cycle must look like ({ai, bi} − {bi, c1−i} − {bi, di} − {di, e1−i} − {di, ai} −{ f1−i, ai} − {ai, bi}). This implies that (a − b − d − a) is a 3-cycle in � contrary to ourassumption.

Suppose � has a nonhex 6-cycle that has two black vertices of type i and one blackvertex of type 1 − i. Such a 6-cycle must look like ({ai, bi} − {bi, c1−i} − {bi, di} −{ei, f1−i} − { f1−i, g1−i} − {h1−i, ki} − {ai, bi}), where e is b or d, h is f or g, and k is aor b. Some of these cases are inconsistent. In each consistent case, one can easily checkthat the girth of � must be less than six. For example, if e = d, h = g, and k = a, then(a − b − d − f − g − a) is a 5-cycle in �.

Therefore, if the girth of � is greater than five, then the only 6-cycles in � are hexes.�

The following proposition is crucial to the proof. Loosely speaking, it allows us toshow that every symmetry of � projects to a symmetry of �.

Proposition 5. For every φ ∈ Aut(�), if [ai]φ = [x j], then [a1−i]φ = [x1− j].

Proof. Let φ be as above and suppose that [ai]φ = [x j]. Let b be a neighbor of ain �. Let y and k be such that [b0]φ = [yk]. Since {ai, b0} is the unique vertex includedin the hexes [ai] and [b0], and [ai]φ = [x j] and [b0]φ = [yk], we can conclude that{ai, b0}φ = {x j, yk}.

We claim that there is a unique element of [b0] that is distance three from the vertex{ai, b0} in �, namely {a1−i, b0}. To see this, first note that certainly none of the otherelements of [b0] are distance three from {ai, b0}. On the other hand, if the distancefrom {ai, b0} to {a1−i, b0} were less than three then the girth of � would be less thansix, which is not the case. Therefore, since φ preserves distance and sends [b0] to [yk],{a1−i, b0}φ is the unique element of [yk] that is distance three from {x j, yk}. That is,{a1−i, b0}φ = {x1− j, yk}.

Now, we know that {a1−i, b0} is a part of exactly two hexes, [a1−i] and [b0]. Also,{x1− j, yk} is a part of exactly two hexes, [x1− j] and [yk]. Since [b0]φ = [yk] and{a1−i, b0}φ = {x1− j, yk}, we can conclude that [a1−i]φ = [x1− j] �

For φ ∈ Aut(�) we now define φ as follows: for a ∈ � and x ∈ �, aφ = x whenever[ai]φ = [x j] for some i, j. By the previous proposition, φ is well defined. It is clear thatφ is a permutation of V (�). We now show that φ ∈ Aut(�).

Suppose that {a, b} is an edge in �. Suppose that {a0, b0}φ = {xi, y j}. By the construc-tion of �, {x, y} must be an edge in �. Notice that {a0, b0} is the unique vertex that ispart of the hexes [a0] and [b0] and that {xi, y j} is part of exactly two hexes: [xi] and [y j].



Therefore, φ sends [a0] and [b0] to [xi] and [y j], in some order. Thus, we may concludethat {a, b}φ = {x, y}. Therefore, φ ∈ Aut(�).

If we let G(φ) = φ, then it is clear that G is a group homomorphism and that for allσ ∈ Aut(�), G(F(σ )) = σ . Notice that β ∈ ker(G).

We now show that � has a color-reversing symmetry if and only if � is bipartite.If � is bipartite, we can define δ: V (�) −→ V (�) as follows: Let S and T be the

partite sets of �. For {ai, b j} ∈ V (�) with a ∈ S and b ∈ T , define {ai, b j}δ = {a1−i, b j}.It is easy to check that since � is bipartite, δ is a symmetry of �. It is clear that δ sodefined is color-reversing and is in the kernel of G. It is also clear that δ commutes withβ. Furthermore, one can easily check that if σ ∈ Aut(�) preserves the partite sets S andT , then σ δ = δσ , while if σ ∈ Aut(�) switches the partite sets S and T , then σ δ = βδσ .

By examining the action of δ in Figure 1, the reader can easily verify the followingproposition.

Proposition 6. If � is bipartite, then for each corner of �, < β, δ > is transitive on theedges of � corresponding to that corner.

In our next proposition, we show that if � is bipartite, then the kernel of G con-sists exactly of the identity, β, δ, and δβ. It easily follows that Aut(�) = {σ β iδ j : σ ∈Aut(�) and i, j ∈ {0, 1}}.Proposition 7. If � is not bipartite then ker(G) = {id, β}. If � is bipartite then ker(G) ={id, β, δ, δβ}.

Proof. From the comments above, we know that {id, β} ⊆ ker(G) and if � is bipartitethen {id, β, δ, δβ} ⊆ ker(G).

Suppose φ ∈ ker(G). It is clear that for each hex [ai], either [ai]φ = [ai] or [ai]φ =[a1−i]. Notice that if [ai]φ = [a j], then [a1−i]φ = [a1− j]. So for each a ∈ � either φ

switches the hexes [a0] and [a1], or φ fixes both the hexes [a0] and [a1].It is not hard to see that if for all [ai], [ai]φ = [ai], then φ must be the identity map.

Indeed, suppose φ fixes each hex. Consider any {ai, b j} ∈ V (�). We know φ sends [ai]to [ai] and [b j] to [b j]. Since {ai, b j} is the unique vertex that is part of [ai] and [b j], itmust be fixed by φ.

Similar reasoning shows that if for all [ai], [ai]φ = [a1−i], then φ must be the map β.Suppose now that φ fixes some hexes and switches others. We will show that this

implies that � is bipartite and that either φ = δ or φ = δβ. Since � is connected, we canfind some edge {a, b} in � so that φ switches [a0] and [a1] while fixing both [b0] and[b1]. It then follows that φ({a0, b0}) = {a1, b0}, so φ is color-reversing. We now colorthe vertices of �. If φ switches the hexes [a0] and [a1], then color a blue. If φ fixes boththe hexes [a0] and [a1], then color a red. We now check that this is a proper coloring.

Suppose {x, y} is an edge in � with x and y having the same color. Consider {x0, y0}, ablack vertex in �. If x and y are both blue then {x0, y0}φ = {x1, y1}. If x and y are both redthen {x0, y0}φ = {x0, y0}. In either case, we contradict the fact that φ is color-reversing.

So the two-coloring we have described is proper and hence � is biparpite. It is easy tocheck that either φ = δ (this happens when the blue vertices are exactly the elements ofS) or φ = δβ (this happens when the blue vertices are exactly the elements of T ). �

Proposition 8. The graph � is 2-arc-transitive if and only if Aut+(�) acts transitivelyon the edges of �.



Proof. Suppose that � is 2-arc-transitive. Consider any two edges e1 ={{ai, bi}, {bi, c1−i}} and e2 = {{x j, y j}, {y j, z1− j}} in �. Find σ ∈ Aut(�) that sends the 2-arc (a − b − c) to the 2-arc (x − y − z). If i = j, then e1σ = e2. If i �= j then e1σ β = e2.In either case, there is a color-preserving symmetry of � which sends e1 to e2.

Conversely, suppose that Aut+(�) acts transitively on the edges of �. Let b ∈ V (�)

with neighbors a, c, and d. Consider the two edges e1 = {{a0, b0}, {b0, c1}} and e2 ={{a0, b0}, {b0, d1}} in �. Suppose φ ∈ Aut+(�) sends e1 to e2. Since φ is color-preserving,we can conclude that φ sends {a0, b0} to {a0, b0} and {b0, c1} to {b0, d1}. Then {a, b}φ ={a, b} and {b, c}φ = {b, d}. Thus, by Proposition 1, � is 2-arc-transitive. �

We now proceed with the proof of Theorem 1.

Proof. Suppose that � is 2-arc-transitive. We know by Proposition 8 that Aut+(�)

acts transitively on the edges of �. If � is bipartite, then δ ∈ Aut(�) is color-reversing;hence, � is symmetric. If, on the other hand, � is not bipartite, then by Proposition 7,every symmetry of � is color-preserving. Hence, � is semisymmetric.

Suppose now that � is not 2-arc-transitive. First notice that if � is not bipartite, thenevery symmetry of � is color-preserving. We know by Proposition 8 that Aut+(�) doesnot act transitively on the edges of �. Hence, � is not edge transitive.

Suppose, finally, that � is bipartite. By Proposition 2, Aut(�) is transitive on corners,and by Proposition 6, < β, δ > is transitive on the edges at each corner, we have thatAut(�) is transitive on all edges. Now, Aut(�) can have no symmetry φ which reversesan edge, for then φ would reverse a corner, proving � to be 2-arc-transitive. �

E. The Dart Graph

III. The Construction. We now define a second construction, this one also producinga tetravalent graph � from a cubic graph �. Let � be a connected, cubic graph. We define� = DG(�), the dart graph of �, in the following way:

Its vertices are the darts (a, b) of �, and two of these vertices are joined by an edgewhen the terminal vertex of one is the initial vertex of the other. We can represent itpictorially by placing the new vertex corresponding to (a, b) on the edge {a, b} in thehalf nearer to b. If the neighbors of b in � are a, c, d, we can picture some of the edgesof � as in Figure 3.

We will use the word hex to mean the set of edges [b] corresponding to a vertex b of�, as in Figure 3.

IV. Results for Dart Graphs of Cubic Graphs. We first make three observations:

(1) � is connected and tetravalent. This is because if {a, b} is an edge of �, then thehexes [a] and [b] have the vertices (a, b) and (b, a) in common, and those verticesare incident with two edges from each of those two hexes.

(2) We can define, as we did in the hexagonal capping of a cubic graph, an injec-tive group homomorphism F : Aut(�) → Aut(�). Here, F(σ ) = σ if (a, b)σ =(aσ, bσ ). Let A be the image of F .

(3) The function β defined by (a, b)β = (b, a) is a symmetry of �. Let P =< A, β >;these are the predicted symmetries of �.

If � has an unpredicted symmetry τ , then for some adjacent hexes [a] and [b], theimages [a]τ and [b]τ must be 6-cycles which are not both hexes but which share opposite



FIGURE 3. Edges in � = DG(�).

vertices. It is not hard to show that this can happen only in the case that � is the cube,Q3. In this case, [Aut(�): P] = 8. The resulting graph is called R12(8, 7) in [13]. It isa dart-transitive graph whose symmetry group has size 768. For all other �, Aut(�) isexactly P.

Proposition 9. � is bipartite if and only if � is bipartite.

Proof. First, suppose that the vertices of � are colored red and blue so that adjacentvertices have different colors. If we assign the color of vertex y to the dart (x, y) for alldarts in �, we see that adjacent vertices in � are assigned different colors.

On the other hand, if � is not bipartite, it contains a cycle (u1 − u2 − · · · − un − u1) ofsome odd length n. Then ((u1, u2) − (u2, u3) − · · · − (un−1, un) − (un, u1) − (u1, u2))

is a cycle in � of the same odd length, and so � is not bipartite. �

Theorem 2. If � is 2-arc-transitive, then � is both semitransitive and dart-transitive.

Proof. Suppose that � is 2-arc transitive. Then the orbit � of the dart ((a, b), (b, c))

under A includes the darts shown in Figure 4, and no others around the vertex b.Then � is a semitransitive orientation of � and so � is semitransitive. Further, β sends

� to its reverse, and so � is dart-transitive. �

Theorem 3. If � is dart-regular, then � is 12 -transitive.

Proof. Suppose that � is dart-regular. Then the orbit of the dart ((a, b), (b, c)) underA includes the darts shown in Figure 5 A, and no others around the vertex b.

Applying β to these gives the darts in Figure 5 B. Thus, the orbit � of ((a, b), (b, c))

under P is a semitransitive orientation of �, and Aut(�) is all of Aut(�). Thus, � is12 -transitive. �

Of the 298 graphs in Foster’s Census, 163 are 2-arc-transitive and so have dart-transitivedart graphs, while the remaining 135 are dart-regular and so have 1

2 -transitive dart graphs.



FIGURE 4. Darts in the orbit of ((a, b), (b, c)) under A.

FIGURE 5. Darts in the orbit of ((a, b), (b, c)) under A and under P.

3. CONSTRUCTIONS FOR MAPS

As we remarked in Section 1, maps and cubic graphs have in common the idea of a corner,though the word is defined differently in these two contexts. This allows us to defineHC(M) (we will continue to call it that, though the Caps are no longer Hexagonal), andDG(M), for a map M. The definitions will be essentially the same, though we will haveto recognize that in a map, more than one edge might have a given pair of endpoints.

The constructed graphs will be coverings of the medial graph MG(M), and we firstwant to mention that several related maps have the same medial graph.

If M is a map, we define the dual of M, denoted D(M), to be a certain map on thesame surface. It has one vertex placed inside each face of M, and one new edge joiningthe two new vertices on opposite sides of each original edge of M; edges and vertices of



M are removed. The new edges divide the surface into faces, one surrounding each ofthe original vertices.

A Petrie path (though it is actually a cycle) in M is a sequence of edges such thateach two consecutive edges are consecutive around some face, but no three consecutivebelong to the same face. Each edge belongs to two Petrie paths. The Petrie of M, denotedP(M), is formed from the underlying graph of M by attaching the boundary of a disk toeach of the Petrie paths. P(P(M)) is isomorphic to M, as is D(D(M)).

By applying these two operators to a map M, we obtain at most six distinct maps:M, P(M), D(M), PD(M), DP(M), and PDP(M) = DPD(M). This last map is usuallyreferred to as the opposite of M, opp(M). For more about these operators, see [15]. It isnot difficult to see that M and all of its related maps have the same medial graph.

F. Cappings of Maps

V. The Construction. We first describe HC(M) where M is a rotary map. The verticesof � = HC(M) are all pairs v = (e, {ai, b j}), where e is an edge of M joining verticesa, b, and i, j ∈ {0, 1}. Edges are all unordered pairs {u, v}, where v = (e, {ai, b j}), u =(e′, {b j, c1−i}), and e, e′ are consecutive edges in some face; i.e., if e, e′ form a corner ofthe map.

In practice, we will often suppress the mention of the edge e, and refer to the vertexsimply as {ai, b j}. With this convention, Figure 1 again describes the constructed graph.

Again the graph is bipartite with partite sets B,W , where B = {{ai, b j}|i = j} andW = {{ai, b j}|i �= j}.

Because the corners of M are the same as the corners of P(M), the graphs HC(M)and HC(P(M)) are isomorphic. However, even though D(M) and opp(M) have thesame medial graph as M, their cappings may be distinct; i.e., the six maps related toM (including M itself, of course) may give as many as three distinct graphs by the HCconstruction.

VI. Results. It is clear that � is a covering of the medial graph, MG(M). In fact,since � is bipartite, it is a cover of B(MG(M)), the bipartite double cover (some saythe canonical double cover ) of MG(M). Thus, if MG(M) is bipartite, its double coveris disconnected, and so the constructed graph is disconnected; in this case, re-defineHC(M) to be one component of the constructed graph. This might be isomorphic toMG(M) or it might be a twofold cover of it. At the moment, it is an open question todetermine when each of these happens.

On the other hand, suppose that MG(M) is not bipartite, and consider a cycle of oddlength starting at some {a, b}. This cycle lifts to a path T in HC(M) starting from a blackvertex {a0, b0} and leading to a white vertex, say {a0, b1}. Because M is rotary, there isa symmetry γ = RS−1 = SR−1 of M which fixes that edge and interchanges a and b.Then T γ is a path from {b0, a0} to {b0, a1}. It follows, then, that � is connected, and sohas eight times as many edges as M does.

We can define, as before, F : Aut(�) → Aut(�). Let A be the image of F . We canalso define β: V (�) −→ V (�) by (e, {ai, b j})β = (e, {a1−i, b1− j}). It is clear that β is asymmetry of �. Let P =< A, β >. These are the predicted symmetries of �.

None of the results concerning the girth of HC(M) carry over from HC(�), and hencewe have no results limiting the size or structure of the symmetry group. In fact, examplesexist which show that the ratio of the sizes of the symmetry groups of � and M can bearbitrarily large.



Many of the results from Section 2.1.2 carry over to HC(M) and some do not. Wepresent here those that do (two as statements, two as theorems); their proofs are essentiallythe same as for HC(�).

(1) If M is reflexible, then P acts transitively on edges of �.(2) If M is bipartite, then � has a color-reversing symmetry. It is unknown whether

the converse holds.

Theorem 4. If M is reflexible, then � =HC(M) is bi-transitive. If, in addition, M isbipartite, then � is dart-transitive.

Remark. IfM is reflexible but not bipartite, � may be semisymmetric or dart-transitive.Examples exist showing that both possibilities actually occur, and it is an unsolvedproblem to tell which reflexible maps have semisymmetric cappings.

Theorem 5. If M is chiral and bipartite, then � =HC(M) is semitransitive.

Remark. The group P need not act transitively on edges of �. If M is bipartite, then� may be dart-transitive or 1

2 -transitive. If M is not bipartite than � may be 12 -transitive

or not edge-transitive at all.There are 111 nontrivial rotary maps with no more than 25 edges. Applying HC to

these yields 36 distinct tetravalent graphs; 30 of them are dart-transitive, four (on 24, 48,60, and 72 vertices) are semisymmetric, one (on 84 vertices) is 1

2 -transitive, and one isnot edge-transitive.

G. Dart Graph of a Map

VII. The Construction. Again suppose that M is a rotary map. We define DG(M)by adapting the definition of DG(�) in a way similar to the adaptation of HC(�) toHC(M).

The vertices of � = DG(M) are all pairs v = (e, (a, b)), where e is an edge of Mjoining vertices a, b. Edges are all unordered pairs {u, v}, where v = (e, (a, b)), u =(e′, (b, c)), and e, e′ form a corner of the map. Thus, the edges of � around a vertex b ofM appear as in Figure 6.

VIII. Pseudo-orientability. To address the question of connectivity in DG(M), wedefine the word pseudo-orientable in a way similar to but slightly different from thatin [16]: A pseudo-orientation of a map is an orientation of the edges of the map suchthat the edges around each face meet head-to-tail at every vertex. For instance, Figure 7shows a pseudo-orientation of the octahedron (considered as a map O on the sphere). Amap M, then, is pseudo-orientable provided that it admits a pseudo-orientation.

A map M is pseudo-orientable if and only if opp(M) is orientable. For instance,opp(O) is a map of type {3, 6} on the torus.

Proposition 10. A map M is pseudo-orientable if and only if DG(M) is not connected.

Proof. First, assume thatM is pseudo-orientable. Then there are two kinds of verticesin DG(M): those that correspond to darts of the orientation, and those that do not. It isclear that vertices of one kind have as neighbors only vertices of the same kind. Therefore,DG(M) falls into two components, each isomorphic to MG(M).

Conversely, suppose that DG(M) is not connected. Since the cycle of edges arounda face corresponds to two cycles in the dart graph, and since the map is connected, we



FIGURE 6. The dart graph of a map.

FIGURE 7. A pseudo-orientation of O.

can see that there are at most two components and each contains one vertex from eachedge of M. Choosing the darts in M corresponding to the vertices in one component ofDG(M), then, gives a pseudo-orientation of M. �

IX. Results. Here, results are similar, but not identical, to results for DG(�). DefineF : Aut(M) → Aut(�) as before and define β ∈ Aut(�) by (e, (a, b))β = (e, (b, a)).Let P be the subgroup of Aut(�) generated by β and the image of F . These are thepredicted symmetries of �. It is not at all clear when DG(M) might have unpredictedsymmetries; many do.

If the underlying graph of M is bipartite, then again assigning the color of vertexy to each dart (e, (x, y)) is clearly a bipartition of DG(M). However, DG(M) can bebipartite even when M itself is not. For example, the underlying graph of M = {4, 4}3,0

is the product of two 3-cycles and so is clearly not bipartite. However, assigning colorwhite to vertices on “horizontal” edges and black to those on “vertical” edges is a properbipartition of MG(M) and so of its cover DG(M).

Theorem 6. If M is reflexible, then � = DG(M) is semitransitive and dart-transitive.

Proof. If M is reflexible, and some corner of a face is made by the edges e1, e2, wherethe endpoints of e1 are a and b, the ends of e2 are b and c, then ((e1, (a, b)), (e2, (b, c))) is



a dart of DG(M), and its orbit � under A includes ((e2, (c, b)), (e1, (b, a))) and neitherof the other two darts at that corner. Then � is a semitransitive orientation of �. Becauseβ sends � to its reverse, � is also dart-transitive. �

Theorem 7. If M is chiral, then � = DG(M) is semitransitive.

Proof. If M is chiral, then the orbit under A of the dart ((e1, (a, b)), (e2, (b, c)))

includes none of the other three darts at that corner. Applying β gives((e1, (b, a)), (e2, (c, b))). Then the orbit under P of that dart is a semitransitive ori-entation for �. If there are no unpredicted symmetries, then � is 1

2 -transitive. �

If we consider the 36 chiral maps with no more than 60 edges, we see that 19 of thecorresponding dart graphs are disconnected, each part being isomorphic to the medialgraph. Of those 19 medial graphs, 10 are dart-transitive, and 9 are 1

2 -transitive. Of course,medial graphs of chiral maps are known to be a rich source of 1

2 -transitive graphs.See [5], for example. The dart graphs of the remaining 17 maps are connected; 12 aredart-transitive, 5 are 1

2 -transitive.There are duplications among these; only 16 different graphs are generated. Seven of

them are 12 -transitive: they have orders 27, 39, 54, 55, 55, 57, and 60.

ACKNOWLEDGMENTS

The research presented in this article was begun, and the principal part completed, as partof the Research Experience for Undergraduates at Northern Arizona University duringthe summer of 2004, supported by National Science Foundation grant DMS-0139523.

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four constructions of highly symmetric tetravalent graphs

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