Semiregular automorphisms of cubic vertex-transitive graphs

We characterise connected cubic graphs admitting a vertex- transitive group of automorphisms with an abelian normal subgroup that is not semiregular. We illustrate the utility of this result by using it to prove that the order of a semiregular subgroup of maximum order in a vertex-transitive group of automorphisms of a connected cubic graph grows with the order of the graph.


Introduction
All the graphs and groups considered in this paper are finite.A very useful tool in the theory of group actions on graphs is the normal quotient method (NQM).This is used to study (and possibly classify) a family of pairs (Γ, G) having certain additional properties, where Γ is a finite graph and G is a subgroup of the automorphism group Aut(Γ) of Γ. (For example, the family consisting of the pairs (Γ, G) where Γ is a finite G-vertex-transitive graph.)The NQM has an impressive pedigree (for example, see [5,6,11,13,14]).To use this method, one generally splits the analysis into two cases, as follows: 1. every nontrivial normal subgroup of G is transitive; 2. G has a nontrivial intransitive normal subgroup N .
In case 1, G is a quasiprimitive group.These groups are classified (see [12]) and have a very restricted structure.In many applications, this allows this case to be completely dealt with.The difficulty usually lies in case 2. Here, one usually considers the quotient pair (Γ/N, G/N ) (see Section 1.1 for the definition).Typically, this pair still lies in the family under consideration and, since Γ/N is smaller than Γ, it is natural to try to use an inductive approach.However, it is often very difficult to recover information about (Γ, G) from (Γ/N, G/N ).
We now describe a variant of the NQM which is sometimes more successful, the abelian normal quotient method (ANQM).As before, one starts with a family of pairs (Γ, G) to study but the analysis is usually split into three cases: 1. G has no nontrivial abelian normal subgroups; 2. G has an abelian normal subgroup that is not semiregular; 3. every abelian normal subgroup of G is semiregular and G has at least one such subgroup, say N .
The main advantage of the ANQM over the NQM lies in case 3. Just as in case 2 of the NQM, one usually considers the quotient pair (Γ/N, G/N ), but the fact that N is abelian and semiregular is often of tremendous help.
The comparative disadvantage is that cases 1 and 2 of the ANQM are potentially more difficult than case 1 of the NQM.For some problems this is an advantageous trade-off and many recent papers have used this approach (see for example [4,7,9,17,19]).
We now explain why cases 1 and 2 of the ANQM are often manageable.In case 1, G has trivial soluble radical.Such a group has some well-known properties: its socle is a direct product of nonabelian simple groups and the group acts faithfully on its socle by conjugation.In particular, the Classification of Finite Simple Groups can be brought to bear on the problem to obtain very detailed information.
Similarly, the situation in case 2 is surprisingly restrictive and very strong results can often be proved under this hypothesis.Consider, for example, the following theorem due to Praeger and Xu (the graphs which appear in the statement will be defined in Section 2): the electronic journal of combinatorics 22(3) (2015), #P3.32 Theorem 1.1 ([15, Theorem 1]).Let Λ be a connected 4-valent G-arc-transitive graph.If G has an abelian normal subgroup that is not semiregular then Λ ∼ = PX(2, r, s) for some r 3 and 1 s r − 1.
Clearly, Theorem 1.1 is very useful when applying the abelian normal quotient method to 4-valent arc-transitive graphs, as it deals with case 2 as satisfactorily as one could hope for, that is, giving a complete classification of the possible graphs.(For examples of applications, see [9,17,18].) One of our goals is to prove the following analogue of Theorem 1.1 for cubic vertextransitive graphs (the graphs which appear in Theorem 1.2 will be defined in Section 2): Theorem 1.2.Let Γ be a connected cubic G-vertex-transitive graph.If G has an abelian normal subgroup that is not semiregular then Γ is isomorphic to one of K 4 , K 3,3 , Q 3 or SPX(2, r, s) for some r 3 and 1 s r − 1.
Much like Theorem 1.1 with respect to 4-valent arc-transitive graphs, Theorem 1.2 will be very useful when applying the abelian normal quotient method to cubic vertextransitive graphs.To illustrate this usefulness, we prove the following: 12] was recently shown to be false by the second author [16].Note also that Theorem 1.3 has appeared previously in [7], however the proof in that paper contains a critical mistake (in the proof of Claim 2, on the last page).
In our proof of Theorem 1.3 we do not make any effort to optimise or even keep track of the most rapidly growing function f satisfying the hypothesis.Our current proof shows that f (n) can be taken to be log(log(n)).However, we conjecture that this is far from best possible: Conjecture 1.4.There exists a constant c > 0 such that in Theorem 1.3 we can take In some sense, Conjecture 1.4 is best possible as it was shown in [3] that f (n) n 1/3 , for infinitely many values of n.

Notation and structure of the paper
The notation used throughout this paper is standard.If Γ is a graph and u and v are adjacent vertices of Γ then u and v are neighbours of each other and (u, v) is an arc of Γ.The set of neighbours of v is called its neighbourhood and is denoted by Γ(v).
If G Aut(Γ), we say that Γ is G-vertex-transitive (respectively, G-arc-transitive) if G acts transitively on the vertices (respectively, arcs) of Γ.The stabiliser of the vertex v in G is denoted by G v and G Γ(v) v denotes the permutation group induced by G v in its action on Γ(v).
Let Γ be a G-vertex-transitive graph and let N be a normal subgroup of G.For every vertex v, the N -orbit containing v is denoted by v N .The normal quotient graph Γ/N has the N -orbits on V(Γ) as vertices, with an edge between distinct vertices v N and w N if and only if there is an edge of Γ between v and w , for some v ∈ v N and some w ∈ w N .Note that G has an induced transitive action on the vertices of Γ/N .Moreover, it is easily seen that the valency of Γ/N is less or equal to the valency of Γ.
The dihedral group of order 2r is denoted by D r .It is usually viewed as a permutation group on the set Z r in a natural way.We also identify the regular cyclic subgroup of D r with Z r .
The remainder of our paper is divided as follows: in Section 2, we define the graphs which appear in Theorems 1.1 and 1.2, prove some useful results about them, and prove Theorem 1.2.Theorem 1.3 is proved in Section 3.

Praeger-Xu graphs and their split graphs
We first define the graphs PX(2, r, s) and prove some useful results about them.Definition 2.1.Let r and s be positive integers with r 3 and 1 s r − 1.The graph PX(2, r, s) has vertex-set Z s 2 ×Z r and edge-set Here is another description of these graphs that is more geometric and sometimes easier to work with.First, the graph PX(2, r, 1) is the lexicographic product of a cycle of length r and an edgeless graph on two vertices.In other words, V(PX(2, r, 1)) = Z 2 × Z r with (u, x) being adjacent to (v, y) if and only if x − y ∈ {−1, 1}.Next, a path in PX(2, r, 1) is called traversing if it contains at most one vertex from Z 2 × {y}, for each y ∈ Z r .Finally, for s 2, the graph PX(2, r, s) has vertex-set the set of traversing paths of PX(2, r, 1) of length s − 1, with two such paths being adjacent in PX(2, r, s) if and only if their union is a traversing path of length s in PX(2, r, 1).
It is not hard to see that this is equivalent to the original definition and that PX(2, r, s) is a connected 4-valent graph with r2 s vertices.Observe that there is a natural action of the wreath product W := Z 2 wr D r = Z r 2 D r as a group of automorphisms of PX(2, r, 1) with an induced faithful arc-transitive action on PX(2, r, s), for every s ∈ {1, . . ., r − 1}.Specifically, W acts on V(PX(2, r, s)) = Z s 2 × Z r in the following way: for g = (g 0 , . . ., g r−1 , h) ∈ W (with g 0 , . . ., g r−1 ∈ Z 2 and h ∈ D r ), we have where the subscripts are taken modulo r and x h denotes the image of x under h.We will also need the concept of an arc-transitive cycle decomposition, which was studied in some detail in [8].
Note that Split(Λ, C) is a cubic graph.We now consider a very important cycle decomposition of PX(2, r, s): 2 , let x ∈ Z r and let C n,x be the cycle of length four of PX(2, r, s) given by ((0, n, x), (n, 0, x + 1), (1, n, x), (n, 1, x + 1)).It is not hard to see that the graph SPX(2, r, s) can also be described in the following way: its vertex-set is Z s 2 × Z r × {+, −} and its edge-set is It is clear from this definition that the graph SPX(2, r, s) is bipartite.Note also that if one contracts every edge of the form {(n, x, −), (n, x, +)} in SPX(2, r, s), one recovers PX(2, r, s).
Observe that the wreath product Namely, for g = (g 0 , . . ., g r−1 , h) ∈ W (with g 0 , . . ., g s−1 ∈ Z 2 and h ∈ D r ), we have where the subscripts are taken modulo r and x h denotes the image of x under h.It is easy to check that W is a vertex-transitive group of automorphisms of SPX(2, r, s).
The graphs SPX(2, r, s) have appeared before in the literature, see for example Suppose first that r = 4.In this case, we actually prove that C is the natural cycle decomposition.By [15, Theorem 2.13], we have Aut(Λ) = W .Let π be the canonical

. , n s−1 ).
There are now four cases to consider.If In all cases we find that C = C n,x .Since C is an arbitrary element of C we have shown that C is the natural cycle decomposition of Λ.
Suppose now that we have |π(C)| 3 for some C ∈ C. In particular, C contains a 2-path P such that π(P ) = (x, x + 1, x + 2) for some x ∈ Z r .Since C is preserved by an arc-transitive group of automorphisms of Λ, there exists g ∈ Aut(Λ) such that g acts on C as a one-step rotation.As Aut(Λ) = W , we have g = (g 0 , . . ., g r−1 , h), for some g 0 , . . ., g r−1 ∈ Z 2 and h ∈ D r .Up to replacing g by its inverse, we may assume that π(P g ) = (x + 1, x + 2, x + 3).In particular, h has order r.Since C is a 4-cycle and r = 4, this is a contradiction.
If r = 4 then 1 s 3 and there are only three graphs to consider: PX(2, 4, 1), PX(2, 4, 2) and PX (2,4,3).The statement can then be checked case-by-case, either by hand or with the assistance of a computer.We now introduce another construction which is, in some sense, an inverse to Construction 2.3 (see Theorem 2.7).

Construction 2.6 ([10, Construction 7]). The input of this construction is a pair (
The output is a decomposition of the edge-set of Γ into a perfect matching T (Γ, G) and a union of cycles R(Γ, G), as well as a graph M(Γ, G) and a partition C(Γ, G) of the edges of M(Γ, G).
Clearly, G v fixes exactly one neighbour of v, and hence each vertex u ∈ V(Γ) has a unique neighbour (which we will denote u ) with the property that G u = G u .Observe that, for every g ∈ G and every v ∈ V(Γ), we have v = v and (v ) g = (v g ) .It follows that the set T (Γ, G) := {{v, v } : v ∈ V(Γ)} is a G-edge-orbit forming a perfect matching of Γ.
We define a new graph M(Γ, G), with vertex-set T (Γ, G) and two elements {u, u } and {v, v } of T (Γ, G) adjacent if and only if there is an edge in Γ between {u, u } and {v, v }; that is, if and only if there is a member of {u, u } adjacent to a member of {v, v } in Γ.
Furthermore, since G is vertex-transitive and G v has two orbits on Γ(v) (one of them being {v } and the other one being Γ(v) \ {v }), G has exactly two arc-orbits, and, since G is not edge-transitive, G also has exactly two edge-orbits (one of them being T (Γ, G)).Since T (Γ, G) forms a perfect matching, the other edge-orbit (which we will call R(Γ, G)) induces a subgraph isomorphic to a disjoint union of cycles, say C 1 , . . ., C n .Finally, let ι be the map A circular ladder graph is the Cartesian product of a cycle of length at least 3 with a complete graph on 2 vertices.When n 2, the Cayley graph Cay(Z 2n , {1, −1, n}) is called a Möbius ladder graph.
We collect a few results about Construction 2.6 which were proved in [10].
Theorem 2.7 ([10, Lemma 9, Theorems 10 and 12]).Let Γ be a connected cubic Gvertex-transitive graph such that G Let K 4 denote the complete graph on 4 vertices, K 3,3 the complete bipartite graph with parts of size 3 and Q 3 the 3-cube.We now prove Theorem 1.2, which we restate for convenience.
Theorem 1.2.Let Γ be a connected cubic G-vertex-transitive graph.If G has an abelian normal subgroup that is not semiregular then Γ is isomorphic to one of K 4 , K 3,3 , Q 3 or SPX(2, r, s) for some r 3 and 1 s r − 1.
Proof.Let v ∈ V(Γ), let N be an abelian normal subgroup of G that is not semiregular and let p be a prime dividing |N v |.Note that the subgroup of N generated by the elements of order p is elementary abelian, is not semiregular and is characteristic in N , and thus normal in G.In particular, replacing N by this subgroup, we may assume that N is an elementary abelian p-group.Note also that, as N is abelian and not semiregular, N is intransitive.Furthermore, since Γ is cubic and connected, G v is a {2, 3}-group, and hence p ∈ {2, 3}.
Suppose that p = 3.Since N is not semiregular, we have is transitive, this implies that every neighbour of v has the same neighbourhood.Therefore Γ ∼ = K 3,3 .
the electronic journal of combinatorics 22(3) (2015), #P3.32 Suppose that p = 2. Since N is not semiregular, we have and C(Γ, G) be as in Construction 2.6.Let k be the length of the cycles in R(Γ, G) (and thus also in C(Γ, G)).
Let {u, v} ∈ R(Γ, G), let C be the cycle of Γ−T (Γ, G) containing u and v, and observe that C is a block of imprimitivity for G and hence also for N .Note that N u and N v act on C as reflections fixing adjacent vertices.Therefore N v , N u fixes C setwise, and the permutation group induced by N v , N u on C is either D k (when k is odd) or D k/2 (when k is even).Since N is abelian, it follows that k = 4.
Suppose that Γ is a circular ladder graph of order 2n.If n = 4 then Γ ∼ = Q 3 .We thus assume that n = 4.In particular, some edges are contained in a unique 4-cycle while others are contained in more than one 4-cycle.Call the latter rungs.Since G has two orbits on edges and the rungs form a perfect matching, T (Γ, G) must be the set of rungs.This implies that Γ − T (Γ, G) consists of two cycles of length n, contradicting the fact that k = 4.
Suppose now that Γ is a Möbius ladder graph of order 2n.If n = 2 then Γ ∼ = K 4 and if n = 3 then Γ ∼ = K 3,3 .We thus assume that n 4 and the same argument as in the last paragraph yields again that T (Γ, G) is the set of edges that are contained in more than one 4-cycle.The removal of these leaves a cycle of length 2n, which is a contradiction.
We may thus assume that Γ is neither a circular ladder nor a Möbius ladder graph.By Theorem 2. The remaining results in this section are observations about the automorphism group of SPX(2, r, s).They will be useful in the proof of Theorem 1.3.Proof.Let Γ = SPX(2, r, s), let G = Aut(Γ) and let v be a vertex of Γ.Note that Γ is not arc-transitive: some edges are contained in cycles of length four, others are not.Let W = Z 2 wr D r = Z r 2 D r act on Γ as described in Definition 2.4.Since W G and Let M(Γ, G) be as in Construction 2.6.Then M(Γ, G) ∼ = PX(2, r, s) (see Definition 2.4).Note that not every edge of Γ is contained in a 4-cycle.In particular, Γ is not isomorphic to a circular ladder graph or a Möbius ladder graph.It follows by Theorem 2.7 that G acts faithfully as a group of automorphisms of M(Γ, G), that is, G Aut(M(Γ, G)) ∼ = Aut(PX(2, r, s)).By [15, Theorem 2.13], Aut(PX(2, r, s)) = W and thus W = G.Corollary 2.9.Let r and s be integers satisfying r 5 and 1 s r − 1, and let G be a vertex-transitive group of automorphisms of SPX(2, r, s).Then G contains a semiregular element of order at least r.
Proof.Let Γ = SPX(2, r, s).We use the definition of SPX(2, r, s) from Definition 2.4 so that V(Γ) = Z s 2 × Z r × {+, −}.By Lemma 2.8 we have that Aut(Γ) = Z r 2 D r .From Definition 2.4, we see that the action of Z r 2 D r on V(Γ) induces a regular action of D r on Z r × {+, −}.
Let π : Aut(Γ) → D r be the natural projection.Since G acts transitively on V(Γ), we obtain that G projects surjectively onto D r , that is, π(G) = D r .Therefore, G contains an element g = (g 0 , . . ., g r−1 , h) with g 0 , . . ., g r−1 ∈ Z 2 and h an element of order r in D r .Clearly, g has order a multiple of r and, writing x = g 0 + g If x = 0 then g r = 1 and g is a semiregular element of order r.If x = 1 then g r = (1, . . ., 1, 1) is a semiregular involution and hence g is semiregular of order 2r.
3 Proof of Theorem 1.3 Theorem 1.3.There exists a function f : N → N satisfying f (n) → ∞ as n → ∞ such that, if Γ is a connected G-vertex-transitive cubic graph of order n then G contains a semiregular subgroup of order at least f (n).
Proof.Our proof uses the abelian normal quotient method and Theorem 1.2.We argue by contradiction and hence we begin by assuming that there exists no such function f .This means that there exist a constant c and an infinite family r, s) into cycles of length four called the natural cycle decomposition of PX(2, r, s).As the arc-transitive action of Z 2 wr D r on PX(2, r, s) induces a transitive action on C, we see that C is arctransitive.The graph Split(PX(2, r, s), C) is simply denoted by SPX(2, r, s).

[ 4 ,Lemma 2 . 5 . 2 D
Section 3]  and[9, Corollary 1.5].Up to conjugacy in Aut(PX(2, r, s)), the natural cycle decomposition of PX(2, r, s) is the unique arc-transitive cycle decomposition of PX(2, r, s) into cycles of length four.Proof.Let Λ = PX(2, r, s), let W = Z r r and let C be an arbitrary arc-transitive cycle decomposition of Λ into cycles of length four.We show that C is conjugate to the natural cycle decomposition of Λ under Aut(Λ).
with Γ k a connected G k -vertex-transitive cubic graph, such that sup{|V(Γ k )| | k ∈ N} = ∞ and every semiregular subgroup of G k has order at most c.For every k, let M k be a normal subgroup of G k of maximal cardinality subject to Γ k /M k being cubic and letF * = {(Γ k /M k , G k /M k )} k∈N .Observe that M k coincides with the kernel of the action of G k on M k -orbits and that M k is semiregular.In particular,|M k | c and moreover, if H k /M k is a semiregular subgroup of G k /M k in its action on V(Γ k /M k ), then H k is semiregular.It follows that Γ k /M k is a connected G k /M k -vertextransitive cubic graph such that sup{|V(Γ k /M k )| | k ∈ N} = sup{|V(Γ k )|/|M k | | k ∈ N} = ∞ and every semiregular subgroup of G k /M k has order at most c/|M k | c.Replacing F by F * , we may thus assume that for every nontrivial normal subgroup M k of G k , the normal quotient Γ k /M k has valency less than three.Replacing F by a subfamily, we may also assume that one of the following occurs:the electronic journal of combinatorics 22(3) (2015), #P3.32Since |N k | c, we have |G k : C G k (N k )| |Aut(N k )| c!. Thus |G k /K k : K k C G k (N k )/K k | c!.Recall that G k /K k acts faithfully and vertex-transitively on the cycle Γ k /N k of length k and thus contains a rotation of order at least k /2.Since |G k /K k : K k C G k (N k )/K k | c!, it follows that C G k (N k ) contains an element g k acting on Γ k /N k as a rotation of order r k with r k k /(2c!).Now, g r k k ∈ K k ∩ C G k (N k ) = C K k (N k ) =N k and hence g r k k is semiregular.Since g k acts semiregularly on Γ k /N k , it follows that g k is semiregular.In particular, g k is a semiregular subgroup of G k of order at least r k k /(2c!) |V(Γ k )|/(2cc!).Since sup{|V(Γ k )| | k ∈ N} = ∞, this is our final contradiction.
transitive cubic graph of order n then G contains a semiregular subgroup of order at least f (n).
Definition 2.2.A cycle in a graph is a connected regular subgraph of valency 2. A cycle decomposition C of a graph Λ is a set of cycles in Λ such that each edge of Λ belongs to exactly one cycle in C. If there exists an arc-transitive group of automorphisms of Λ that maps every cycle of C to a cycle in C then C will be called arc-transitive.Construction 2.3 ([10, Construction 11]).The input of this construction is a pair (Λ, C), where Λ is a 4-valent graph and C is an arc-transitive cycle decomposition of Λ.The output is the graph Split(Λ, C), the vertices of which are the pairs (v, C) where v ∈ V(Λ), C ∈ C and v lies on the cycle C, and two vertices (v 1 , C 1 ) and (v 2 , C 2 ) are adjacent if and only if either