Rigidity and Symmetry pp 27-40 | Cite as

# Cubic Cayley Graphs and Snarks

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## Abstract

The well-known conjecture that there are no snarks amongst Cayley graphs is considered. Combining the theory of Cayley maps with the existence of certain kinds of independent sets of vertices in arc-transitive graphs, some new partial results are obtained suggesting promising future research directions in regards to this conjecture.

## Keywords

Snark Cubic graph Cayley graph Coloring Arc-transitive Independent set## Subject Classifications

05C25 05C15 20E32## 1 Introductory Remarks

A *snark* is a connected, cyclically 4-edge-connected cubic graph which is not 3-edge-colorable, that is, a connected, cyclically 4-edge-connected cubic graph whose edges cannot be colored by three colors in such a way that adjacent edges receive distinct colors. While examples of snarks were initially scarce – the Petersen graph being the first known example of a snark – infinite families of snarks are now known to exist. The first and second Blanuša snarks, the second and third snarks discovered [5], are actually the smallest members of two infinite families of snarks [31].

A *Cayley snark* is a cubic Cayley graph which is a snark. Although most known examples of snarks exhibit a lot of symmetry, none of them is a Cayley graph. In fact it was conjectured in [3] that no such graphs exist.

### Conjecture 1 ([3]).

There are no Cayley snarks.

The proof of this conjecture would contribute significantly to various open problems regarding Cayley graphs. One of such problems is the well-known conjecture that every connected Cayley graph contains a Hamilton cycle. Namely, every hamiltonian cubic graph is easily seen to be 3-edge-colorable. It is perhaps also worth mentioning that Conjecture 1 is in fact a special case of the conjecture that all Cayley graphs on groups of even order are 1-factorizable (see [32]). (A graph is *1-factorizable* if its edge set can be partitioned into edge-disjoint 1-factors (perfect matchings).)

A large number of articles, directly or indirectly related to this problem, have appeared in the literature affirming the non-existence of Cayley snarks. For example, in [28] it is proved that the smallest example of a Cayley snark, if it exists, comes either from a non-abelian simple group or from a group which has a single non-trivial proper normal subgroup. The subgroup must have index two and must be either non-abelian simple or the direct product of two isomorphic non-abelian simple groups. In 2004, Potočnik [30], motivated by the fact that there are only two known examples of connected cubic vertex-transitive graphs which are not 3-edge-colorable, namely, the Petersen graph and its truncation, asked whether every connected cubic vertex-transitive graph, other than these two graphs, is 3-edge-colorable, and gave the answer for graphs admitting transitive solvable groups of automorphisms. In particular, it is proved in [30] that every connected cubic graph (different from the Petersen graph) whose automorphism group contains a solvable subgroup acting transitively on the set of vertices is 3-edge-colorable.

In this paper we will present an innovative approach to finding a possible solution to Conjecture 1 combining the theory of Cayley maps with the existence of a certain kind of independent set of vertices in arc-transitive graphs (see Theorem 1).

The paper is organized as follows. In Sect. 2 we gather various concepts that are needed in the subsequent sections. In Sect. 3 we discuss the structure of Cayley snarks. In Sect. 4 we describe the above mentioned approach and then use it to prove Conjecture 1 for a certain class of graphs in Sect. 5. Finally, in Sect. 6 we give possible future directions in regards to the Cayley snark problem.

## 2 Terminology and Notation

Throughout this paper graphs are simple and, unless otherwise specified, undirected and connected. Furthermore, all graphs and groups are assumed to be finite. For group-theoretic terms not defined here we refer the reader to [34].

Let *X* be a graph. Then for adjacent vertices *u* and *v* in *X*, we write *u* ∼ *v* and denote the corresponding edge by *uv*. We let *V* (*X*), *E*(*X*), *A*(*X*) and Aut*X* be the vertex set, the edge set, the arc set and the automorphism group of *X*, respectively. If *u* ∈ *V* (*X*) then *N* _{ X }(*u*) denotes the set of neighbors of *u*. The *girth* of *X* is the length of a shortest cycle in *X*. For a non-negative integer *k*, a *k-arc* in *X* is a sequence of *k* + 1 vertices (*u* _{1}, *u* _{2}, *…*, *u* _{ k+1}), not necessarily all distinct, such that any two consecutive terms are adjacent and any three consecutive terms are distinct. For a subset *U* of *V* (*X*) the subgraph of *X* induced by *U* is denoted by *X*[*U*]. If *X*[*U*] is an empty graph then *U* is called an *independent set of vertices*.

A *k-factor* of a graph is a spanning *k*-regular subgraph of the graph. Therefore a 2-factor is a collection of cycles spanning all vertices of the graph. A 2-factor is said to be *even* if all of these cycles are of even length. A *Hamilton path* of a graph is a simple path going through all vertices of the graph. A *Hamilton cycle* of a graph is a cycle going through all vertices of the graph; in other words, it is a connected 2-factor of a graph. A graph possessing a Hamilton cycle is said to be *hamiltonian*.

A subgroup *G* ≤ Aut*X* is said to be *vertex-transitive*, *edge-transitive* and *arc-transitive* provided it acts transitively on the set of vertices, edges and arcs of *X*, respectively. A graph is said to be *vertex-transitive*, *edge-transitive*, and *arc-transitive* if its automorphism group is vertex-transitive, edge-transitive and arc-transitive, respectively. A subgroup *G* ≤ Aut*X* is said to be *k-regular* if it acts transitively on the set of *k*-arcs and the stabilizer of a *k*-arc in *G* is trivial. A graph *X* is said to be *(G,k)-regular* if *G* ≤ Aut*X* is *k*-regular. In particular, a subgroup *G* ≤ Aut*X* is said to be 1-*regular* if it acts transitively on the set of arcs and the stabilizer of an arc in *G* is trivial.

*G*and a subset \(S \subseteq G\) such that \(S = {S}^{-1}\) and 1 ∉

*S*, the

*Cayley graph*Cay(

*G*,

*S*) on

*G*relative to

*S*has vertex set

*G*and edge set {

*g*∼

*gs*∣

*g*∈

*G*,

*s*∈

*S*}. If

*G*is cyclic then Cay(

*G*,

*S*) is said to be a

*circulant*. Note that Cay(

*G*,

*S*) is connected if and only if

*S*is a generating set of the group

*G*. Denote by Aut(

*G*,

*S*) the set of all automorphisms of a group

*G*which fix the set

*S*⊆

*G*setwise, that is,

It is easy to check that Aut(*G*, *S*) is a subgroup of Aut(Cay(*G*, *S*)) contained in the stabilizer of the identity element 1 ∈ *G*. It follows from the definition of Cayley graphs that the left regular representation *G* _{ L } of *G* induces a regular subgroup of Aut(Cay(*G*, *S*)), implying that Cay(*G*, *S*) is a vertex-transitive graph. Following [38], Cay(*G*, *S*) is called a *normal Cayley graph* if *G* _{ L } is normal in Aut(Cay(*G*, *S*)), that is, if Aut(*G*, *S*) coincides with the vertex stabilizer 1 ∈ *G*. Moreover, if Cay(*G*, *S*) is a normal Cayley graph, then Aut(Cay(*G*, *S*)) = *G* _{ L } *⋊* Aut(*G*, *S*).

In order to state the classification of connected arc-transitive circulants, which has been obtained independently by Kovács [19] and Li [23], we need to recall certain graph products. The *wreath (lexicographic) product X*[*Y* ] of a graph *X* by a graph *Y* is the graph with vertex set *V* (*X*) × *V* (*Y* ) such that {(*u* _{1}, *u* _{2}), (*v* _{1}, *v* _{2})} is an edge if and only if either {*u* _{1}, *v* _{1}} ∈ *E*(*X*), or *u* _{1} = *v* _{1} and {*u* _{2}, *v* _{2}} ∈ *E*(*Y* ). For a positive integer *b* and a graph *X*, denote by *bX* the graph consisting of *b* vertex-disjoint copies of the graph *X*. Then the graph \(X[\overline{K_{b}}] - bX\) is called the *deleted wreath (deleted lexicographic) product* of *X* and \(\overline{K_{b}}\), where \(\overline{K_{b}} = bK_{1}\).

### Proposition 1 ([19, 23]).

*If X is a connected arc-transitive circulant of order n, then one of the following holds:*

- (i)
*X ≅ K*_{ n }; - (ii)
\(X =\varSigma [\overline{K}_{d}],\)

*where n = md, m,d > 1 and Σ is a connected arc-transitive circulant of order m;* - (iii)
\(X =\varSigma [\overline{K}_{d}] - d\varSigma,\)

*where*\(n = md,\ d > 3,\ gcd(d,m) = 1\)*and Σ is a connected arc-transitive circulant of order m;* - (iv)
*X is a normal circulant.*

Near-bipartite graphs are a natural generalization of bipartite graphs. A *near-bipartite* graph is a graph *X* in which there exists an independent set *I* ⊂ *V* (*X*) of vertices, such that the induced graph *X*[*V* (*X*)*∖ I*] is bipartite. A *chromatic number χ*(*X*) of a graph *X* is the minimum number of colors needed to color the vertices of *X* in such a way that adjacent vertices have different colors. If the graph is near-bipartite, than we can color the vertices of this graph with three colors, since we can color the vertices in the independent set with one color, and for the remaining vertices two colors suffice. Therefore, near-bipartite graphs have chromatic number at most 3. Conversely, if a graph has chromatic number 3, then it is near-bipartite, since we can choose the vertices of one color to be the required independent set, and the remaining vertices are colored with two colors, which means that the remaining graph is bipartite.

The following result about chromatic numbers of tetravalent circulants will be needed in Sect. 4.

### Proposition 2 ([16, Theorem 3.2.]).

*Let*\(X = C_{n}(a,b) = \text{Cay}(\mathbb{Z}_{n},\{\pm a,\pm b\})\)

*be a tetravalent circulant of order n. Then*

Following [29] we say that, given a graph (or more generally a loopless multigraph) *X*, a subset *S* of *V* (*X*) is *cyclically stable* if the induced subgraph *X*[*S*] is acyclic, that is, a forest. The size | *S* | of a maximum cyclically stable subset *S* of *V* (*X*) is called the *cyclic stability number* of *X*.

Given a connected graph *X*, a subset *F* ⊆ *E*(*X*) of edges of *X* is said to be *cycle-separating* if *X* − *F* is disconnected and at least two of its components contain cycles. We say that *X* is *cyclically k-edge-connected* if no set of fewer than *k* edges is cycle-separating in *X*. Furthermore, the *edge-cyclic connectivity ζ*(*X*) of *X* is the largest integer *k* not exceeding the Betti number \(\vert E(X)\vert -\vert V (X)\vert + 1\) of *X* for which *X* is cyclically *k*-edge-connected. (This distinction is indeed necessary as, for example, the theta graph *Θ* _{2}, *K* _{4} and *K* _{3,3} possess no cycle-separating sets of edges and are thus cyclically *k*-edge-connected for all *k*, however their edge-cyclic connectivities are 2, 3 and 4, respectively.)

Regarding the cyclic stability number, the following result may be deduced from [29, Théorème 5].

### Proposition 3 ([29]).

*Let X be a cyclically 4-edge-connected cubic graph of order n, and let S be a maximum cyclically stable subset of V (X). Then*\(\vert S\vert = \lfloor (3n - 2)/4\rfloor \)

*and more precisely, the following hold.*

- (i)
*If*\(n \equiv 2\pmod 4\)*, then*\(\vert S\vert = (3n - 2)/4\)*, and X[S] is a tree and V (X)∖S is an independent set of vertices.* - (ii)
*If*\(n \equiv 0\pmod 4\)*, then*\(\vert S\vert = (3n - 4)/4\)*, and either X[S] is a tree and V (X)∖S induces a graph with a single edge, or X[S] has two components and V (X)∖S is an independent set of vertices.*

The following result concerning cyclic edge connectivity of cubic vertex-transitive graphs is proved in [27, Theorem 17].

### Proposition 4 ([27]).

*The cyclic edge connectivity ζ(X) of a cubic connected vertex-transitive graph X equals its girth g(X).*

## 3 On Cayley (Non)Snarks

Any cubic Cayley graph is bridgeless, and so, by Petersen’s theorem [33], it has a 1-factor and a 2-factor. In [20] the existence of 2-factors with long cycles in cubic graphs is investigated, whereas, in this paper we are after even 2-factors. Namely, a cubic graph containing such a 2-factor is not a snark for we can color the edges of an even 2-factor with two colors and the remaining edges with the third color.

For a cubic Cayley graph Cay(*G*, *S*) the generating set *S* is of two forms: either it consists of three involutions or it consists of an involution, a non-involution and its inverse. In the first case, Cay(*G*, *S*) is clearly 3-edge-colorable, and we may therefore restrict ourselves to the study of cubic Cayley graphs with respect to generating sets with a single involution. More precisely, we consider Cayley graphs *X* arising from groups having a (2, *s*, *t*)-generation, that is, from groups \(G =\langle a,x\mid {a}^{2} = {x}^{s} = {(ax)}^{t} = 1,\ldots \rangle\) generated by an involution *a* and an element *x* of order *s* ≥ 3 such that their product *ax* has order *t*. Here “*…*” denotes the extra relations needed in the presentation of the group. Such graphs are called *(2,s,t)-Cayley graphs*. If *s* is even, then a (2, *s*, *t*)-Cayley graph is not a snark, since in this case the set of edges {{*g*, *gx*}∣*g* ∈ *G*} obviously forms an even 2-factor in *X*. The following proposition thus holds.

### Proposition 5.

*A Cayley snark is a (2,s,t)-Cayley graph, where s is odd, not possessing an even 2-factor.*

As cubic graphs are of even order, by Proposition 5, the existence of a Hamilton cycle in a cubic Cayley graph implies that the graph is 3-edge-colorable, and thus a non-snark. In particular, Conjecture 1 is essentially a weaker form of the folklore conjecture that every connected Cayley graph with order greater than 2 possesses a Hamilton cycle, which is in fact a Cayley variation of Lovász’s conjecture [24] that every connected vertex-transitive graph possesses a Hamilton path. These hamiltonicity questions have been challenging mathematicians for more than 40 years, but only partial results have been obtained thus far. Most results proved thus far depend on various restrictions made either on the class or the order of the group or the structure of the corresponding generating sets. For example, one may easily see that connected Cayley graphs on abelian groups have a Hamilton cycle. Further, it is known that the same holds for hamiltonian groups (see [2]), for metacyclic groups with respect to standard generating sets (see [1]), and for groups with a cyclic commutator subgroup of prime-power order (see [9, 18, 25, 35]). This last result was generalized to connected vertex-transitive graphs admitting a transitive group of automorphisms with a cyclic commutator subgroup of prime-power order, where the Petersen graph is the only exception (see [8]). In addition, every connected Cayley digraph on any *p*-group has a directed Hamilton cycle (see [36]). On the other hand, it is still not known whether Cayley graphs on dihedral groups have a Hamilton cycle. The best result in this respect is due to Alspach, Chen and Dean [4] who solved the problem for generalized dihedral groups of order divisible by 4. Furthermore, combining results from [10, 11, 22, 26] it follows that every connected Cayley graph on a group *G* has a Hamilton cycle if | *G* | = *kp*, where *p* is prime, 1 ≤ *k* < 32, and *k* ≠ 24. These results show, among other, that every connected Cayley graph on a group of order *n* < 120, *n* ≠ 72, is hamiltonian. Moreover, since every group of order 72 is solvable, these results and [30, Theorem 1.5.] combined together imply the following proposition.

For further results not explicitly mentioned or referred to here see the survey articles [7, 21, 37].

## 4 Constructing Even 2-Factors in (2, *s*, *t*)-Cayley Graphs

The methods used in this paper to construct even 2-factors in cubic Cayley graphs are a generalization of the methods introduced by Glover and Yang in [13]. These methods were later used in [12, 14, 15] where the hamiltonian problem for (2, *s*, 3)-Cayley graphs was considered.

*s*≥ 3 and

*t*≥ 3 be positive integers and let \(X = \text{Cay}(G,\{a,x,{x}^{-1}\})\) be a (2,

*s*,

*t*)-Cayley graph on a group \(G =\langle a,x\mid {a}^{2} = {x}^{s} = {(ax)}^{t} = 1,\ldots \rangle\). The graph

*X*is cubic and has a canonical Cayley map \(\mathcal{M}(X)\) given by an embedding in the closed orientable surface of genus

*G*| ∕

*s*disjoint

*s*-gons and |

*G*| ∕

*t*2

*t*-gons as the corresponding faces. (For a detailed description of Cayley maps we refer the reader to [6, 17].) This map is given using the same rotation of the

*x*,

*a*,

*x*

^{−1}edges at every vertex and results in one

*s*-gon and two 2

*t*-gons adjacent to each vertex. Generalizing the approach of [12] where the so called hexagon graph was associated with each (2,

*s*, 3)-Cayley graph, a

*2t-gonal graph O*(

*X*) is associated with our (2,

*s*,

*t*)-Cayley graph

*X*in the following way. Its vertex set consists of all 2

*t*-gons in \(\mathcal{M}(X)\) arising from the relation (

*ax*)

^{ t }= 1, where two such 2

*t*-gons are adjacent if they share an edge in

*X*. Observe that

*O*(

*X*) may also be seen as the

*orbital graph*of the left action of

*G*on the set \(\mathcal{C}\) of left cosets of the subgroup \(C =\langle ax\rangle\), arising from the suborbit {

*aC*,

*caC*,

*c*

^{2}

*aC*,

*c*

^{3}

*aC*,

*… c*

^{ t−1}

*aC*} of length

*t*, where

*c*=

*ax*. More precisely,

*O*(

*X*) has vertex set \(\mathcal{C}\), with adjacency defined as follows: a coset

*yC*is adjacent to the

*t*cosets

*yaC*,

*ycaC*,

*yc*

^{2}

*aC*,

*…*,

*yc*

^{ t−2}

*aC*and

*yc*

^{ t−1}

*aC*. Clearly,

*G*acts 1-regularly on

*O*(

*X*). Conversely, let

*Y*be a connected arc-transitive graph of valency

*t*admitting a 1-regular action of a subgroup

*G*of Aut

*Y*. Let

*v*∈

*V*(

*Y*) and let

*h*be a generator of \(C = G_{v}\mathop{\cong}\mathbb{Z}_{t}\). Then there must exist an element

*a*∈

*G*such that \(G =\langle a,h\rangle\) and such that

*Y*is isomorphic to the orbital graph of

*G*relative to the suborbit {

*aC*,

*haC*,

*h*

^{2}

*aC*,

*…*,

*h*

^{ t−1}

*aC*}. Moreover, a short computation shows that

*a*may be chosen to be an involution, and letting

*x*=

*ah*we get the desired generation for

*G*. There is therefore a well-defined correspondence between these two classes of objects. This gives us the following result.

### Proposition 7.

*Let X be a (G,1)-regular graph of valency t, t ≥ 3, with the vertex stabilizer G* _{v} *isomorphic to* \(\mathbb{Z}_{t}\) *. Then X can be constructed via a Cayley graph on the group G with respect to its (2,s,t)-generation. In particular, X is isomorphic to a 2t-gonal graph of a (2,s,t)-Cayley graph on the group G.*

The method for constructing an even 2-factor in *X* consists in identifying a subset *T* of vertices *V* = *V* (*O*(*X*)) inducing a bipartite subgraph in *O*(*X*), the complement *V ∖ T* of which is an independent set of vertices. (Of course the existence of such an even 2-factor is obvious in the case that *s* is even.) The bipartite subgraph *O*(*X*)[*T*] gives rise to a “bipartite graph of faces” in *X* such that every vertex in the Cayley graph *X* lies on the boundary of at least one of the faces in this subgraph. (The concept of a “bipartite graph of faces” is defined in the obvious way, see the examples below.) Since all faces in this graph of faces are of even length (they are 2*t*-gonal faces), its boundary is the desired even 2-factor in the (2, *s*, *t*)-Cayley graph *X*. In particular, this method works in case *O*(*X*) is a near-bipartite graph (see Theorem 1).

### Example 1.

On the right-hand picture of Fig. 1 we show a forest of faces whose boundary is a 2-factor consisting of even cycles in the spherical Cayley map \(\mathcal{M}(X)\) of the Cayley graph *X* on the (2, 5, 3)-generated group \(A_{5} =\langle a,x\mid {a}^{2} = {x}^{5} = {(ax)}^{3} = 1\rangle\). The corresponding forest in *O*(*X*) is shown on the left-hand picture of Fig. 1.

### Example 2.

On the left-hand picture of Fig. 2 we show a graph of faces whose boundary is an even 2-factor in the spherical Cayley map of the Cayley graph on the symmetric group *S* _{4} with respect to its (2, 3, 4)-generation \(\langle a,x\mid {a}^{2} = {x}^{3} = {(ax)}^{4} = 1,\ldots \rangle\), where *a* = (13) and *x* = (234). The corresponding bipartite induced subgraph of *O*(*X*) is shown on the right-hand picture of Fig. 2.

### Example 3.

Let \(G = \mathbb{Z}_{4} \ltimes (\mathbb{Z}_{3} \times \mathbb{Z}_{3}) =\langle x,y,f\mid {x}^{3} = {y}^{3} = {f}^{4} = 1,{x}^{y} = x,{x}^{f} = y,{y}^{f} = {x}^{-1}\rangle\). Then *a*: = *f* ^{2} *xy* and *b*: = *fy* give rise to a (2, 4, 4)-generation of *G*. Let *X* be a (2, 4, 4)-Cayley graph on *G* with respect to this (2, 4, 4)-generation. Then using (1) it can be seen that *X* has a toroidal Cayley map given by an embedding in the torus with 18 faces, 9 disjoint squares and 9 octagons. By Proposition 7 the corresponding 8-gonal graph *O*(*X*) is an arc-transitive graph of order 9 admitting a 1-regular action of *G* with a vertex stabilizer isomorphic to \(\mathbb{Z}_{4}\). Figure 3 shows that *O*(*X*) ≅ *K* _{3}[3*K* _{1}] − 3*K* _{3} is a near-bipartite graph.

### Theorem 1.

*Let* \(X = \text{Cay}(G,\{a,x,{x}^{-1}\})\) *be a (2,s,t)-Cayley graph on a group* \(G =\langle a,x\mid {a}^{2} = {x}^{s} = {(ax)}^{t} = 1,\ldots \rangle\) *, s,t ≥ 3 and let O(X) be the corresponding 2t-gonal graph of X. If O(X) is a near-bipartite graph, then X is not a snark. Moreover, if the vertex set V of O(X) decomposes into {T,V ∖T} where T induces a tree and V − T is an independent set of vertices, then X contains a Hamilton cycle.*

### Proof.

Consider the canonical Cayley map \(\mathcal{M}(X)\) of \(X = \text{Cay}(G,\{a,x,{x}^{-1}\})\) embedded with *s*-gonal and 2*t*-gonal faces in a closed orientable surface with genus as given in (1). Suppose that the corresponding 2*t*-gonal graph *O*(*X*) is near-bipartite. Then the vertex set *V* of *O*(*X*) decomposes into {*T*, *V ∖ T*} where *T* induces a bipartite graph and *V ∖ T* is an independent set of vertices. Each vertex of *O*(*X*) corresponds to a 2*t*-gonal face of \(\mathcal{M}(X)\) as illustrated in the beginning of this section. Since every vertex of *X* belongs to two 2*t*-gonal faces in \(\mathcal{M}(X)\) and *V ∖ T* is an independent set of vertices, we can conclude that every vertex of *X* belongs to at least one 2*t*-gonal faces whose corresponding vertex of *O*(*X*) is in *T*. The bipartite graph *O*(*X*)[*T*] then translates into a graph of faces in \(\mathcal{M}(X)\) whose boundary contains all the vertices of *X*. Since all faces in this graph of faces are of even length (they are 2*t*-gonal faces), its boundary is an even 2-factor in *X*. By Proposition 5 it now follows that *X* is not a snark. Furthermore, if *O*(*X*)[*T*] is a tree, then it translates into a tree of faces in \(\mathcal{M}(X)\) containing all of the vertices of *X* and as a subspace of the Cayley map it is a topological disk. The boundary of this topological disk is a (simple) cycle passing through all vertices of the Cayley graph *X*, and so *X* is hamiltonian. □

### Theorem 2.

*Let* \(X = \text{Cay}(G,\{a,x,{x}^{-1}\})\) *be a (2,s,4)-Cayley graph on a group* \(G =\langle a,x\mid {a}^{2} = {x}^{s} = {(ax)}^{4} = 1,\ldots \rangle\) *, s ≥ 3, such that the corresponding 8-gonal graph O(X) is a circulant. Then X is not a snark.*

### Proof.

By Proposition 7, the 8-gonal graph *O*(*X*) is a tetravalent (*G*, 1)- regular circulant, and so Proposition 1 implies that *O*(*X*) is either isomorphic to the complete graph *K* _{5}, or to \(K_{2}[5K_{1}] - 5K_{2}\mathop{\cong}K_{5,5} - 5K_{2}\), or to *K* _{2}[4*K* _{1}] ≅ *K* _{4, 4}, or to *C* _{ n }[2*K* _{1}], or it is a normal circulant. Since the order of *O*(*X*) is equal to \(\vert G\vert /4 = \vert V (X)\vert /4\), in the first three possibilities Proposition 6 implies that *X* is not a snark. In what follows we may therefore assume that *O*(*X*) is of order ≥ 30 and we only need to consider two cases.

### Case 1.

*O*(*X*) ≅ *C* _{ n }[2*K* _{1}].

We can clearly color the vertices of *C* _{ n } with three colors (namely, *C* _{ n } is near-bipartite). With the use of such a vertex coloring of *C* _{ n } we can now color the graph *C* _{ n }[2*K* _{1}] in the following way. If a vertex *v* of *C* _{ n } is colored with color *i*, then we color the two vertices corresponding to this vertex (the two vertices in the same 2*K* _{1} corresponding to *v*) with color *i*. This gives us a good 3-vertex coloring of *C* _{ n }[2*K* _{1}], implying that *O*(*X*) is near bipartite, and therefore, by Theorem 1, *X* is not a snark.

### Case 2.

*O*(*X*) is a normal circulant.

In this case \(O(X)\mathop{\cong}\text{Cay}(\mathbb{Z}_{n},S)\) and the stabilizer of the vertex \(0 \in \text{Cay}(\mathbb{Z}_{n},S)\) in the full automorphism group of *O*(*X*) is isomorphic to \(\text{Aut}(\mathbb{Z}_{n},S) \leq \text{Aut}(\mathbb{Z}_{n})\mathop{\cong}\mathbb{Z}_{n}^{{\ast}}\). Since *O*(*X*) is connected and \(\langle S\rangle = \mathbb{Z}_{n}\) we may, without loss of generality, assume that *S* = { ±1, ±1^{ α }} for some \(\alpha \in \text{Aut}(\mathbb{Z}_{n})\). Since *O*(*X*) is of order ≥ 30, Proposition 2 implies that *χ*(*O*(*X*)) ≤ 3. Therefore *O*(*X*) is a near-bipartite graph and thus, by Theorem 1, *X* is not a snark. □

## 5 Existence of Even 2-Factors in (2, *s*, 3)-Cayley Graphs

It was proved in [12] that a (2, *s*, 3)-Cayley graph on a group *G* has a Hamilton path when | *G* | is congruent to 0 modulo 4, and has a Hamilton cycle when | *G* | is congruent to 2 modulo 4. The Hamilton cycle was constructed, combining the theory of Cayley maps with classical results on cyclic stability in cubic graphs, as the contractible boundary of a “tree of faces” in the corresponding Cayley map. Further, with a generalization of these methods, the existence of a Hamilton cycle in a (2, *s*, 3)-Cayley graph was proved in [14] when apart from | *G* | also *s* is congruent to 0 modulo 4. More recently, with a further extension of the above “tree of faces” approach, a Hamilton cycle was shown to exist whenever | *G* | is congruent to 0 modulo 4 and *s* is odd (see [15]). This leaves | *G* | congruent to 0 modulo 4 with *s* congruent to 2 modulo 4 as the only remaining case in which the existence of a Hamilton cycle in (2, *s*, 3)-Cayley graphs has not yet been proven. In this last case, however, the “tree of faces” approach cannot be applied, and so entirely different techniques will have to be introduced if one is to complete the proof of the existence of Hamilton cycles in (2, *s*, 3)-Cayley graphs. These results combined together with Proposition 5 imply that there are no (2, *s*, 3)-Cayley snarks. For the sake of completeness, however, a self-contained proof of this fact is provided below. These demonstrate that it is somewhat easier to deal with the snark problem than the hamiltonian problem, bearing in mind of course that both problems are among the hardest problems in graph theory. The following proposition, which combined together with Proposition 3 shows that cyclically 4-edge-connected cubic graphs are near-bipartite, will be needed in this respect. In fact, the statement of this proposition really says that in part (ii) of Proposition 3, a particular one of the two possibilities for the set *S* may be chosen.

### Proposition 8.

*Let X be a cyclically 4-edge-connected cubic graph of order* \(n \equiv 0\pmod 4\) *. Then there exists a cyclically stable subset S of V (X) such that X[S] is a forest and V (X)∖S is an independent set of vertices.*

### Proof.

Let *X* be a cyclically 4-edge-connected cubic graph of order \(n \equiv 0\pmod 4\). By Proposition 3(ii) we may assume that there exists a maximum cyclically stable subset *S* of *V* (*X*) such that *X*[*S*] is a tree and *V* (*X*)*∖ S* induces a graph with a single edge, say *uv* ∈ *E*(*X*). Consider the neighbors of the vertex *u* ∈ *V* (*X*) in *X*[*S*]. Since \(v \in N_{X}(u) \cap V (X)\setminus S\) is the only neighbor of *u* in *V* (*X*)*∖ S* it follows that | *N* _{ X[S]}(*u*) | = 2. Let \(N_{X[S]}(u) =\{ u_{i}\mid i \in \{ 1,2\}\}\). Then | *N* _{ X[S]}(*u* _{ i }) | ≤ 2, *i* ∈ { 1, 2}. Furthermore, if for some *i* ∈ { 1, 2} we have | *N* _{ X[S]}(*u* _{ i }) | = 2 then the set \(\{u\} \cup S\setminus \{u_{i}\}\) induces a forest whose complement is an independent set of vertices. We may therefore assume that | *N* _{ X[S]}(*u* _{ i }) | = 1 for every *i* ∈ { 1, 2}. Also, since *X*[*S*] is connected there exists a path *P* = *u* _{1} *w* _{1} *w* _{2} *… w* _{ k } *u* _{2} between vertices *u* _{1} and *u* _{2} in *X*[*S*]. If there exists *j* ∈ { 1, *…*, *k*} such that | *N* _{ X[S]}(*w* _{ j }) | = 3, then the set \(\{u\} \cup S\setminus \{w_{j}\}\) induces a forest whose complement is an independent set of vertices. Hence, we may assume that *X*[*S*] = *P*. Now, repeating the argument for the neighbors \(N_{X[S]}(v) =\{ v_{i}\mid i \in \{ 1,2\}\}\) of the vertex *v* in *X*[*S*] it follows that we can restrict ourselves to the case that | *N* _{ X[S]}(*v* _{ i }) | = 1 for every *i* ∈ { 1, 2}. But *u* _{1} and *u* _{2} are the only vertices of valency 1 in *X*[*S*], and so \(\{v_{i}\mid i \in \{ 1,2\}\} =\{ u_{i}\mid i \in \{ 1,2\}\}\). It follows that *uvu* _{ i } *u*, *i* ∈ { 1, 2}, is a 3-cycle in *X*, a contradiction (since *X* is cyclically 4-edge-connected). □

### Theorem 3.

*There are no (2,s,3)-Cayley snarks.*

### Proof.

Let *X* be a (2, *s*, 3)-Cayley graph on a group \(G =\langle a,x\mid {a}^{2} = {x}^{s} = {(ax)}^{3} = 1,\ldots \rangle\), *s* ≥ 3, and let *O*(*X*) be the corresponding 6-gonal graph of *X*. By Proposition 7, *O*(*X*) is a (*G*, 1)-regular graph. By Proposition 6 we may assume that *O*(*X*) is of order | *G* | ∕3 ≥ 60, and therefore [12, Proposition 3.4.] implies that *O*(*X*) is of girth strictly bigger than 5. Proposition 4 implies that *O*(*X*) is a cyclically 4-edge-connected graph, and so Propositions 3 and 8 combined together imply that *O*(*X*) is a near-bipartite graph. Thus, by Theorem 1, *X* is not a snark. □

## 6 Further Research Directions

By Proposition 5 and Theorem 3, a Cayley snark, if it exists, is a (2, *s*, *t*)-Cayley graph on a group \(G =\langle a,x\mid {a}^{2} = {x}^{s} = {(ax)}^{t} = 1,\ldots \rangle\), where *s* ≥ 3 is odd and *t* > 3, which does not have an even 2-factor, and, by Theorem 1, its corresponding 2*t*-gonal graph, arising from the canonical Cayley map \(\mathcal{M}(X)\) given by an embedding of it in the closed orientable surface with *s*-gonal and 2*t*-gonal faces, is not near-bipartite. The converse is not true. In particular, there exist (2, *s*, *t*)-Cayley graphs which are not snarks but their 2*t*-gonal graphs are not near-bipartite. For example, the 2*t*-gonal graph arising from the spherical Cayley map of a (2, 3, 3)-Cayley graph *X* on the alternating group \(A_{4} =\langle a,x\mid {a}^{2} = {x}^{3} = {(ax)}^{3} = 1\rangle\), where *a* = (12)(34) and *x* = (123), is isomorphic to the complete graph *K* _{4}, and it is therefore not near-bipartite. By Proposition 6, however, *X* is not a snark.

In view of Proposition 7 the 2*t*-gonal graph *O*(*X*) of a (2, *s*, *t*)-Cayley graph *X* on a group \(G =\langle a,x\mid {a}^{2} = {x}^{s} = {(ax)}^{t} = 1,\ldots \rangle\), *s*, *t* ≥ 3, is a (*G*, 1)-regular graph of valency *t* with the vertex stabilizer *G* _{ v } isomorphic to \(\mathbb{Z}_{t}\). It therefore seems that the thoughtful study of the structure of such graphs is in order if one is to make a progress in regards to Conjecture 1. We pose the following problem.

### Problem 1.

Let *G* be a finite group. Characterize non-near-bipartite *t*-valent (*G*, 1)-regular graphs with the vertex stabilizer *G* _{ v } isomorphic to \(\mathbb{Z}_{t}\).

Since a graph is near-bipartite if and only if its vertices can be colored with less than four colors, Problem 1 can be reformulated as follows.

### Problem 2.

Let *G* be a finite group. Characterize *t*-valent (*G*, 1)-regular graphs *X* with the vertex stabilizer *G* _{ v } isomorphic to \(\mathbb{Z}_{t}\) such that *χ*(*X*) ≥ 4.

## Notes

### Acknowledgements

The first author was supported in part by ARRS, P1-0285 and research project “mladi raziskovalci”. The second author was supported in part by ARRS, P1-0285, J1-2055 and Z1-4006, and by ESF EuroGiga GReGAS. The third author was supported in part by ARRS, P1-0285, J1-2055, J1-4010 and J1-4021, and by ESF EuroGiga GReGAS.

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