Cubic Vertex-Transitive Graphs Admitting Automorphisms of Large Order

A connected graph of order n admitting a semiregular automorphism of order n/k is called a k-multicirculant. Highly symmetric multicirculants of small valency have been extensively studied, and several classification results exist for cubic vertex- and arc-transitive multicirculants. In this paper, we study the broader class of cubic vertex-transitive graphs of order n admitting an automorphism of order n/3 or larger that may not be semiregular. In particular, we show that any such graph is either a k-multicirculant for some k≤3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \le 3$$\end{document}, or it belongs to an infinite family of graphs of girth 6.


Introduction
Studying the structure of the automorphism groups of highly symmetrical graphs is one of the classical topics in the area of algebraic graph theory and a very important part of it aims at proving upper bounds on the order of the automorphism groups in terms of a conveniently tame function of the order of the graph (see, for example, the classical work of Tutte [30] on cubic symmetric graphs).Existence of such bounds often allows strong group theoretical tools to be applied.
In some applications (such as a construction of a complete list of all graphs of fixed valence, bounded order and given symmetry type; see for example [6,21]), a bound on the order of the automorphism group can be substituted with a weaker result where the order of individual automorphisms are bounded rather than the order of the automorphism group itself.It is well known that the order of the automorphism group of a connected vertex-transitive graph of valence 3 (cubic vertex-transitive graph, for short) cannot be bounded by any subexponential function of the order (see, for example, [21]).However, a recent result [25,Theorem 1.6] shows that the order o(g) of an individual automorphism g of a cubic vertex-transitive graph other than K 3,3 equals the length (g) of the longest orbit of the cyclic group g , implying that o(g) cannot exceed the order of the graph.In other words, if we define η(Γ) := |V(Γ)| max{o(g) : g ∈ Aut(Γ)} , then η(Γ) ≥ 1 holds for every cubic vertex-transitive graph.The aim of this paper is to investigate how sharp this bound is and under what additional assumptions it can be impoved.More precisely, we obtain a complete classification of cubic vertex-transitive graphs Γ for which 1 ≤ η(Γ) ≤ 3 holds and thus show that η(Γ) > 3 for every cubic vertex-transitive graph not belonging to a known list of exceptional families (see Theorem 1.1).Before stating Theorem 1.1, let us first introduce a few notions appearing in its statement.A nontrivial automorphism g of a graph is called semiregular provided that the length of every vertex-orbit of g equals o(g), or equivalently, when all the orbits of g have equal size.A graph admitting a semiregular automorphism with k vertex-orbits is called a k-multicirculant; in addition, every graph on n vertices is an n-multicirculant.Let κ(Γ) := min{k : Γ is a k-multicirciulant} and observe that η(Γ) ≤ κ(Γ) ≤ |V(Γ)| holds for every graph Γ.What is more, the well-known polycirculant conjecture [14] (which is known to be true for graphs of valence 3 [15], as well as many other classes of graphs; see [1]) states that κ(Γ) < |V(Γ)| for every vertex-transitive graph Γ.There are numerous classification results proved about cubic vertex-transitive k-multicirculants of different symmetry types (see for example [5,8,10,11,12,13]) and in particular, all cubic vertex-transitive 1-multicirculants (also called circulants), 2-multicirculant (also called bicirculants) and 3-multicirculants (also called triciculants) are known [20,22].
An interesting interplay between the parameters η(Γ) and κ(Γ) is considered in Section 8.In particular, the question whether the parameter κ(Γ) can be bounded above in terms of η(Γ) is discusses there.
Let us now introduce some families of graphs appearing in Theorem 1.1.For a positive integer n ≥ 3, let Prism(n) denote the prism on 2n vertices, which can also be viewed as the generalised Petersen graph GP(n, 1), and let Moeb(n), n ≥ 4 even, be the Möbius ladder with vertex-set Z m and edges of the form {x, x + s} for x ∈ Z n , s ∈ {±1, n/2}.Note that κ(Moeb(n)) = 1 for every n, while κ(Prism(n)) is 1 or 2, depending on whether n is odd or even, respectively.
Further, for a positive integer n and distinct elements i, j ∈ Z n \ {0}, let H(n, i, j) be the graph with vertex-set Z n × Z 2 and edges of the form {(x, 0), (x + s, 1)} for x ∈ Z n and s ∈ {0, i, j}.Note that the graphs H(n, i, j) are bipartite bicirculants and are also known as cyclic Haar graphs [20].Clearly, κ(H(n, i, j)) ≤ 2 and the values of n, i, j for which κ(H(n, i, j)) = 1 are characterised in Lemma 7.1.
Let us now define two families of tricirculants, first introduced in [22, Definitions 4.1 and 5.1].For an odd integer k, k > 1, let X(k) be the graph with 6k vertices, labelled u i , v i and w i for i ∈ Z 2k , and the edgeset being the union , and E 5 = {{v i , w i+r } : i ∈ Z 2k }, where r = (k + 3)/2 if k ≡ 1 (mod 4) and r = (k + 3)/2 + k if k ≡ 3 (mod 4).
For positive integers m, t ≥ 3, let SDW(m, t) be the cubic graph with the vertex-set Z m × Z t × Z 2 and edges of the form {(x, i, 0), (x, i ± 1, 1)} and {(x, i, 1), (x + 1, i, 0)} for all x ∈ Z m and i ∈ Z t .(The graphs SDW(m, t) with t = 3 are also known as split depleted wreath graphs.)Note that a graph SDW(m, t) is a 2t-multicirculant, as witnessed by the semiregular automorphism of order m, mapping a vertex (x, i, j) to the vertex (x + 1, i, j) for every x, i, j.It can be easily seen that κ(SDW(m, 3)) ≤ 6 and that η(SDW(m, 3)) ≤ 3 unless m is divisible by 6 (in which case η(SDW(m, 3)) = 6); see Lemma 6.13 for more details.
Finally, Tutte's 8-cage is the unique cubic arc-transitive graph on 30 vertices, appearing under the name CubicVTgraph (30,8) in [21], while the truncated tetrahedron is the graph on 12 vertices obtained by the geometric truncation of the skeleton of the tetrahedron (it appears under the name CubicVTgraph (12,2) in [21]).
We can now state the main result of this paper.
Theorem 1.1.Let Γ be a finite simple connected vertex-transitive graph of valence 3 and order n.Then Γ admits an automorphism of order at least n 3 (or equivalently, η(Γ) ≤ 3) if and only if one of the following happens: The proof of the above theorem is inevitably rather technical since a certain amount of case-by-case analysis cannot be avoided.However, we have tried to use as many theoretical tools as possible in order to shorten and organise the arguments into self contained parts.
Our proof relies on two crucial ideas.The first idea is to classify the cubic vertex-transitive graphs Γ admitting an automorphism g of order at least |V(Γ)|/3 in terms of the quotient graphs Γ/ g .For this approach to be practical, we need to store enough information that will allow us to reconstruct the graphs Γ from their quotients.Here, a recently developed theory of generalised cyclic covering projections [23], briefly summarised in Section 3, provided the needed theoretical background.The second crucial fact making this approach feasible follows from the results of [25], from which one can deduce that a cyclic group of automorphisms of a cubic vertex-transitive graph Γ with order at least |V(Γ)| 3 can have at most 5 orbits on V(Γ) (see Lemma 4.1).In particular, there is only a finite number of possible quotients Γ/ g with o(g) ≥ |V(Γ)| 3 .The rest of the proof is then a careful case-by-case analysis of these possible quotient graphs.The strategy of proving Theorem 1.1 is laid out in more details in Section 2.3.

Overview and basic definitions
The following paragraphs, in which we give some basic formal definitions and we outline the proof of Theorem 1.1, serve as a more detailed summary of the contents of this paper.
2.1.Graphs.We would first like to stress that all the graphs in this paper are finite.Even though we are primarily interested in simple graphs (that can be defined as a finite set of vertices together with an irreflexive symmetric relation on it), it will be very convenient for us to adopt a more general definition of a graph that has become standard when quotients and covers of graphs are considered (see, for example, [16]).
For us, a graph is an ordered 4-tuple (V, D; beg, inv) where D and V = ∅ are disjoint finite sets of darts and vertices, respectively, beg : D → V is a mapping which assigns to each dart x its initial vertex beg x, and inv is an involutory permutation of D which interchanges every dart x with its inverse dart, also denoted by x −1 .The final vertex of a dart x is beg x −1 and is denoted end x.The neighbourhood of a vertex v is defined as the set of darts that have v for its initial vertex and the valence of v is the cardinality of the neighbourhood.If Γ is a graph we write V(Γ) and D(Γ) to denote the vertex-and dart-set of Γ, respectively.Furthermore, we may write beg Γ and inv Γ , with a subscript, to indicate the beginning and inverse functions of Γ, to avoid confusion when more than one graph is involved.We will generally omit the subscript if there is no possibility of ambiguity.
The orbits of inv are called edges.The edge {x, and is called a link otherwise.The endvertices of an edge are the initial vertices of the darts contained in the edge.If {u, v} is the set of the endvertices of an edge, then we say that u and v are adjacent and write u ∼ v. Two darts x and y are parallel if beg x = beg y and beg x −1 = beg y −1 .Two edges are parallel if they have the same endvertices.When we present a graph as a drawing, the links are drawn in the usual way as a line between the points representing its endvertices, a loops is drawn as a closed curve at its unique endvertex and a semiedge is drawn as a segment attached to its unique endvertex.
A graph without loops, semiedges and pairs of parallel edges is simple.Note that a simple graph is completely determined by its vertex-set and the adjacency relation, and conversely, given a set V and an irreflexive symmetric relation ∼ on V , we can define a graph (V, D; beg, inv) by letting D := {(u, v) : u, v ∈ V(Γ), u ∼ v}, beg(u, v) = u and inv(u, v) = (v, u).A dart in a simple graph is traditionally called an arc, so we will use these two terms interchangeably.In this paper, a cubic graph will always stand for a connected simple graph in which every vertex has valence 3.
Notions such as morphism, isomorphism and automorphism of graphs are obvious generalisations of those defined in the traditional setting and precise definitions can be found in [16,17].In particular, an automorphism of a graph (V, D; beg, inv) is a permutation g of V ∪ D preserving each of V and D such that inv(x g ) = inv(x) g and beg(x g ) = beg(x) g for every x ∈ D. Note that when the graph is simple, an automorphism is uniquely defined by its adjacency preserving action on the vertex-set.We shall thus often consider view automorphism of simple graphs in the usual way, that is, as adjacency preserving permutations of the vertex-set.

2.2.
Labelled quotients of graphs.Let Γ := (V, D; beg, inv) be a graph admitting a cyclic group of automorphisms G ≤ Aut(Γ).For v ∈ V , let v G denote the G-orbit of v and let V /G be the set of all G-orbits of vertices of Γ.Similarly, let x G be the G-orbit of x ∈ D and let D/G be the set of all G-orbits on darts.We define the G-quotient of Γ as the graph Γ/G = (V /G, D/G, beg , inv ) where beg x G = (beg x) G and inv x G = (inv x) G for all x ∈ D(Γ).
Let x ∈ D(Γ) be a dart and let u = beg x.Then, x G is a dart of Γ/G with initial vertex u G .Let λ G (x G ) denote the number of darts of Γ in the orbit x G that begin at any fixed vertex in u G (note that this is independent of which vertex of u G we choose and that λ G (x G ) = |x Gu |).Then λ G : D(Γ/G) → N is a well-defined function and the pair (Γ/G, λ G ) is called a labelled quotient (see Figure 2.1).

Strategy.
In what follows, we describe our plan to prove Theorem 1.1.We first consult the census of all cubic vertex-transitive graphs [21] and check that the theorem holds for all the graphs on at most The Möbius ladder on 8 vertices (top left) and K 3,3 (top right) above their labelled quotients by a cyclic group of automorphisms G = g .For the case on the left, g is the automorphisms given by the permutation (u 0 u 1 u 2 u 3 )(v 0 v 1 v 2 v 3 ); for the graph on the right g is given by (u 0 u 1 )(v 0 v 1 v 2 ).In the graph on the lower right, the edge joining u G 0 and v G 0 has a label next to each of its endvertices: 3 next to u G 0 , and 2 next to v G 0 .These are the labels λ G (x) and λ G (x −1 ) where x is the dart pointing from u 0 to v 0 .This indicates that every vertex in the fibre of u G 0 (in the graph above) has 3 neighbours in the fibre of v G 0 while every vertex in the fibre of v G 0 has 2 neighbours in the fibre of u G 0 .
20 vertices (this can be easily done with a help of computer and a computer algebra system such as Sage [29]).We may thus concentrate on the class G of all cubic vertex-transitive graphs on at leats n > 20 admitting a cyclic group G ≤ Aut(Γ) of order at least n 3 .Let Q be the set of labelled quotients (Γ/G, λ G ) where Γ ∈ G and G ≤ Aut(Γ) is cyclic of order at least |V(Γ)|

3
(by [25,Theorem 4.7], the order of G equals the order of the largest G-orbit on vertices).
Observe that if (Γ/G, λ G ) ∈ Q, then the graph Γ/G is connected but may admit parallel edges, loops or semi-edges.Since the valence of every vertex in Γ is 3, it follows that the vertices in Γ/G have valence at most 3 and that λ G (x) ≤ 3 for every dart x of Γ/G.Moreover, a labelled graph in Q can have at most 5 vertices (see Lemma 4.1).This shows that the set Q is a subset of the set Q • of all connected subcubic labelled graphs (Q, λ) on at most 5 vertices with λ(x) ≤ 3. The set Q • is clearly finite, but still consists of an inconveniently large number of labelled graphs.
To determine which of the labelled graph in Q • indeed arise as quotients of the cubic graphs in G by an appropriate cyclic group G (that is, which of them belong to Q), we rely on the concept of a cyclic generalised voltage graph, first introduced in [23], and the associated generalised covering graph construction.Loosely speaking, this construction takes a labelled graph (∆, λ), together with an additional information, called voltage assignment ζ, as an input and constructs a connected graph Γ (called a cover, for short) having (∆, λ) as a labelled quotient.It was proved in [23], that as the voltage assignment ζ varies this procedure yields all possible connected graphs having (∆, λ) as a quotient by a cyclic group.
In Section 4, we use the results proved in [23] (and summarised in Section 3) to find a set of necessary conditions (see Theorem 4.2) for a labelled graph in Q • to admit a connected vertex-transitive generalised cyclic cover belonging to G.In Section 5 we determine, by means of forbidden labelled subgraphs, further necessary conditions for an element of Q • to admit a cyclic generalised cover in G (see Theorem 5.9).The set of conditions given in Theorems 4.2 and 5.9 is restrictive enough to allow us to compute, via a brute-force algorithm, the subset Q * ⊆ Q • of labelled graphs satisfying them.There are 20 such graphs.
In Section 6, we analyse the 20 labelled graphs of Q * in detail, and show that precisely nine of them admit vertex-transitive cubic cyclic generalised covers belonging to G (see Figure 2.2).These are the nine elements of the set Q.However, due to some overlap in the families of covering graphs, only seven elements of Q are necessary to reconstruct G. Finally in Section 7, we characterise the elements of G and complete the proof of Theorem 1.1.With the exception of the right-most graph, these graphs correspond to the 8 possible quotients of a k-multicirculant graph by a k-multicirculant automorphism, with k ∈ {1, 2, 3}.

Covers
We now formally introduce the concept of a cyclic generalised voltage graph, which generalises voltage graphs (in the sense of [9]) for cyclic voltage groups, and is a special case of the wider class of generalised voltage graphs introduced in [24].The definitions and results in this section are mostly taken from [23], where cyclic generalised voltage graphs were first defined.Each cyclic generalised voltage graph gives rise to a unique generalised covering graph (called covering graph for simplicity).By the end of the section, we characterise those cyclic generalised voltage graphs whose covering graphs are cubic (that is, connected, finite, simple 3-valent graphs).Definition 3.1.Let ∆ be a finite connected graph and let λ : D(∆) → N, ι : V(∆) → N and ζ : D(∆) → Z be functions such that λ(x)ι(beg x) = λ(x −1 )ι(beg x −1 ), (3.1) for every dart x ∈ D(∆).Then we say that the quadruple (∆, λ, ι, ζ) is a cyclic generalised voltage graph, and we call the functions λ, ι and ζ a labelling, an index function and a voltage assignment, respectively.Definition 3.2.Let (∆, λ, ι, ζ) be a cyclic generalised voltage graph.The cover of (∆, λ, ι, ζ), denoted Cov(∆, λ, ι, ζ) is the graph Γ := (V , D , beg , inv ) where: Remark 3.3.Since the second coordinate of a vertex (v, i) or a dart (x, i) in the definition above is an element of {0, . . ., ι(v) − 1} or {0, . . ., λ(x)ι(beg x) − 1}, respectively, any operation on the second coordinate is to be computed modulo ι(v) or λ(x)ι(beg x) accordingly.In particular, on the right-hand side of equality (4), the sum i + ζ(x) is to be computed modulo λ(x)ι(beg x).
Let (∆, λ, ι, ζ) be a cyclic generalised voltage graph and set Γ = Cov(∆, λ, ι, ζ).For the sake of simplicity, we will write v i instead of (v, i) for a vertex of Γ, and x i instead of (x, i) for a dart of Γ.The natural projection π : Γ → ∆ that maps every v i ∈ V(Γ) to v and every x i ∈ D(Γ) to x is a graph epimorphism.For each vertex v ∈ V(∆) we call the set π −1 (v) = {v i : i ∈ {1, . . ., ι(v)}} the fibre of v and we denote it fib(v).Similarly, the fibre of a dart Let n = lcm{λ(x)ι(beg x) : x ∈ D(∆)} and observe that the group Z n acts on D(Γ) ∪ V(Γ) by the rule The permutation induced by each g ∈ Z n is an automorphism of Γ and the orbit of a dart (or a vertex) z i under this action is precisely fib(z).Furthermore, the action of Z n is faithful and hence there is an embedding φ : Z n → Aut(Γ) (see [23,Lemma 6.3]).The image φ(Z n ) is a cyclic group of order n generated by the automorphism of Γ that maps every vertex v i ∈ V(Γ) to v i+1 and every dart x i ∈ D(Γ) to x i+1 .We call this automorphism the canonical covering transformation of Cov(∆, λ, ι, ζ) and we denote it by ρ.
In short, the covering graph Γ admits a cyclic group of automorphisms of order n whose orbits on vertices and darts are precisely the fibres of vertices and darts of (∆, λ, ι, ζ).
Conversely, in view of [23,Theorem 5.3] and [23, Theorem 6.2] every graph admitting a cyclic subgroup of automorphism of order n is the cover of some cyclic generalised voltage graph (∆, λ, ι, ζ), and by [23,Lemma 6.1], it follows that lcm{λ(x)ι(beg x) : x ∈ D(∆)} = n.Moreover, by [23,Theorem 6.6] we can always assume that ζ(x) = 0 for all darts x lying on a prescribed spanning tree T of ∆.A voltage assignment satisfying this condition is said to be T -normalised.Let us summarise these observations in the following theorem:  ( (3) For every u i ∈ fib(u), there are exactly λ(x) darts in fib(x) that begin at u i ; (4) For every where Since cubic graphs are the main object of study of this paper, we would like to focus precisely on those cyclic generalised voltage graphs whose covers are cubic graphs.Hence the following definition.The following characterization of ccv-graphs is a consequence of [23, Theorems 6.8 and 6.9].Lemma 3.8.Let (∆, λ, ι, ζ) be a cyclic generalised voltage graph where ζ is T -normalised for some spanning tree T of ∆.
3.1.Extendability to ccv-graphs.Let ∆ be a connected finite graph, and let λ : D(∆) → N be an arbitrary function.We call the pair (∆, λ) a labelled graph.Naturally, we obtain a labelled graph (∆, λ) from every cyclic generalised voltage graph (∆, λ, ι, ζ) by simply disregarding the functions ι and ζ.In this case, the fact that λ comes from a generalised voltage graph, which by definition satisfies equality (3.1), restricts λ to some degree.We say that a labelled graph (∆, λ) is extendable if there exist functions ι and ζ such that (∆, λ, ι, ζ) is a cyclic generalised voltage graph.We will be particularly interested in those extendable labelled graphs that can be extended to a ccv-graph.
A walk of length n is a sequence of darts W = (x 1 , x 2 , . . ., x n ) such that beg x i+1 = end x i for all i ∈ {1, . . ., n − 1}.We say W is closed if end x n = beg x 1 , and we say it is reduced if A path is a reduced walk where beg x i = beg x j for all i = j and a cycle is a closed path.A cycle of length n is also called an n-cycle.A tree is a connected graph without any cycles.
For a walk W = (x 1 , x 2 , . . ., x n ), we define the inverse of W as the walk . ., y m ) are two walks such that end x n = beg y 1 , then we define the concatenation of W 1 and W 2 as W 1 W 2 = (x 1 , . . ., x n , y 1 , . . ., y m ).
Let (∆, λ) be a labelled graph.We can extended the labelling λ to a function λ * that assigns to each walk of ∆ a rational number.For a walk We then have that for any two walks W 1 and W 2 for which the concatenation is defined.The following is a useful characterization of labelled graphs that are extendable.Lemma 3.9.[23, Lemma 3.5] A labelled graph (∆, λ) is extendable if and only if λ * (C) = 1 for every closed walk C of ∆.

Simplified voltages.
As one would expect, for given a labelled graph (∆, λ), there may exist different ccv-extension (∆, λ, ι, ζ) and (∆, λ, ι, ζ ) such that Cov(∆, λ, ι, ζ) ∼ = Cov(∆, λ, ι, ζ ).It would be convenient to take, among all the possible ccv-extensions yielding isomorphic covers, one with a voltage assignment that is as "nice" as possible.As was proved in [23,Lemma 7.4], for every ccv-extension (∆, λ, ι, ζ) of a labelled graph (∆, λ) there exists a voltage assignment ζ that is simplified in the sense of Definition 3.11 below, such that Cov(∆, λ, ι, ζ) Remark 3.12.One of the advantages of considering ccv-graphs with simplified voltage assignments is that the adjacency rules for the corresponding covering graphs become quite straightforward.Indeed, suppose (∆, λ, ι, ζ) is a ccv-graph where ζ is simplified.Let x ∈ D(∆), let u = beg x and v = end x, and let e = {x, x −1 }.Then for all u i ∈ fib(u) we have: ( Let (∆, λ, ι, ζ) be a ccv-graph.By [23,Proposition 7.4] there exists a simplified voltage assignment ζ for ∆ such that Cov(∆, λ, ι, ζ) ∼ = Cov(∆, λ, ι, ζ ).This implies that every cubic graph is the cover of a ccv-graph with a simplified voltage assignment.For this reason, we will henceforth always assume that the voltage of a ccv-graph is simplified.The following theorem summarises the contents of this section.Theorem 3.13.A graph Γ is cubic and admits a cyclic group of automorphisms of order n if and only if it is the cover of a cyclic generalised voltage graph (∆, λ, ι, ζ) where λ :

Vertex Transitive Covers
We have shown in Section 3 that if Γ is a cubic graph admitting a cyclic group of automorphisms G, then Γ is isomorphic to a ccv-cover of some labelled graph (∆, λ), where the labelling λ satisfies conditions (1)-( 5) of Lemma 3.10; that is, (∆, λ) is extendable, λ(x) = 1 for every dart x underlying a loop, a semiedge or a link that is parallel to another link, and deg λ (x) = 3 for all darts x ∈ D(∆).If in addition we suppose that Γ is vertex-transitive and that G has order at least V(Γ)/3, then further restrictions are set on the labelled graph (∆, λ).It was shown in [25] that if Γ is a cubic vertex-transitive graph and G ≤ Aut(Γ) is cyclic, then the number of orbits of G is bounded by a function of k = |V(Γ)|/|G|.This, in turn, bounds the number of vertices of the quotient Γ/G.Furthermore, the ratio between the sizes of the largest and smallest orbits of G is also bounded, which restricts the labelling λ.Let us be more precise.
Let Γ be a cubic graph of order n > 20 admitting a cyclic subgroup of automorphisms G. Then Γ is isomorphic to the cover of some ccv-graph (∆, λ, ι, ζ) where lcm{λ(x)ι(beg x) : x ∈ D(∆)} = |G|.We can slightly abuse the language and identify Γ with Cov(∆, λ, ι, ζ).Suppose Γ is vertex-transitive and let m = |G|.Then by [25,Theorem 4.7], a G-orbit on vertices must have size m/i for some i ∈ {1, 2, 3, 4, 6} and the largest G-orbit has size precisely m.It follows that for any two vertices ū and v of Γ, we have Since the orbits of G on V(Γ) are identified with the fibres of V(∆) we see that If in addition we suppose that m ≥ n/3, then there must exist a vertex û ∈ V(∆) such that Furthermore, [25,Theorem 1.6] asserts that if Γ has order n > 20 and ū ∈ Γ is such that ūG is of maximal order among all G-orbits, then ū has a neighbour v such that ūG = vG but |ū G | = |v G |.In particular, since the largest orbit of G has size at least n/3, this implies that at least two thirds of the vertices of Γ are contained in only two orbits (ū G and vG ).It is an easy exercise to see that the remaining vertices can be divided in, at most, 3 different orbits (of size n/9 each, or all three having different sizes: n/6, n/9 and n/18).Hence we obtain the following lemma.

Artefacts
A labelled subgraph of a labelled graph (∆, λ) is a pair (∆ , λ ) where ∆ is a subgraph of ∆ and λ is the restriction λ | D(∆ ) .In this section we will define a set of 'forbidden subgraph' for a labelled graph, that we call artefacts.An artefact in a labelled graph (∆, λ) is a labelled subgraph of (∆, λ) that guaranties the existence of a particular subgraph (containing a short cycle) in any ccv-cover of (∆, λ).We will show that a labelled graph containing certain artefacts cannot admit a vertex-transitive ccv-cover of order large than 10.First, we will define the notion of the signature of a graph.
Let Γ be a cubic graph, let x be a dart of Γ and c be a positive integer.Denote by c (x) the number of c-cycles (cycles of length c) that pass through x.Let v ∈ V(Γ) and let {x 1 , x 2 , x 3 } be the set of darts beginning at v, ordered in such a way that c (x 1 ) ≤ c (x 2 ) ≤ c (x 3 ).The triplet ( c (x 1 ), c (x 2 ), c (x 3 )) is then called the c-signature of v. Informally, the c-signature of v tells us how the cycles of length c passing through v are distributed among the darts incident to v. If all vertices of Γ have the same c-signature, we say that Γ is c-cycle-regular, and we say the c-signature of Γ is the c-signature of any of its vertices.Observe that if Γ is vertex-transitive, then Γ is c-cycle-regular for all c ∈ N.For cubic graphs of small girth g, the g-signature is sometimes enough to completely determine the graph.The following lemma is a direct consequence of the results proved in [26] (or independently in [7]) and [27] .) Let Γ be a cubic girth-regular graph of girth g ≤ 6.Then either the g-signature of Γ is (0, 1, 1) or one of the following occurs: (1) g = 3 and Similarly, the graph SDW(m, 3) was denoted in [27] as Σ m and defined as the Cayley graph on the group D m × Z 3 with respect to the connection set S = {(ρτ, 0), (τ, 1), (τ, 2)}), where D m = ρ, τ | ρ m , τ 2 , (ρτ ) 2 denotes the dihedral group of order 2m.To prove that Σ m is indeed isomorphic to SDW(m, 3) one can check that the mapping (ρ i τ j , k) → (i, k, 1 − j) is indeed a graph isomorphism.As was proved in [27, Proposition 5], the automorphism group of SDW(m, 3), m > 3, is isomorphic to the group S 3 × D m (where by S 3 we denote the symmetric group of order 6).
Corollary 5.3.If Γ is a cubic arc-transitive graph of girth smaller than 6, then Γ is isomorphic to one of the following: K 4 , K 3,3 , the three-dimensional cube Q 3 , the Petersen Graph or the dodecahedron GP(10, 2).Proof.Let u = beg x and u 0 ∈ fib(u).Let a and b be two arbitrary darts of Γ.Since Γ is vertex-transitive, there exist automorphisms φ, ψ ∈ Aut(Γ) such that (beg a) φ = u 0 = (beg b) ψ .In particular, both a φ and b ψ begin at u 0 .Furthermore, since λ(x) = 3, all three darts beginning at u 0 belong to the same orbit of the cyclic group of automorphisms of Γ preserving the fibres, and thus there exists γ ∈ Aut(Γ) such that a φγ = b ψ .Then a φγψ −1 = b.We conclude Γ is arc-transitive.
Proof.The proof consists in repeatedly applying Remark 3.12 for each of the five possible cases.For instance, suppose Γ ∼ = Cov(∆, λ, ι, ζ) where (∆, λ, ι, ζ) is a ccv-extension of (∆, λ) and that Γ contains a subgraph isomorphic to A 1 .Assume the notation of Figure 5.1.Then ι(u) = 2 • ι(v) = 2m for some m ∈ N and since uv is a [1, 2]-edge, it follows from Remark 3.12 that each v i ∈ fib(v) is adjacent to u i and u i+m .Furthermore, since u is incident to a semi-edge, u i is incident to u i+m (again, by Remark 3.12).Then, (u i , u i+m , v i ) is a 3-cycle of Γ for all i ∈ {0, . . ., m − 1}.A similar argument shows that the lemma holds for the four remaining cases (see Figure 5.1).
Let A 6 and A 7 be the labelled graphs depicted in the bottom row of Figure 5.2.Proof.Let (∆, λ, ι, ζ) be a ccv-extension of (∆, λ) such that Γ ∼ = Cov(∆, λ, ι, ζ).Suppose, for a contradiction, that Γ is vertex-transitive and (∆, λ) contains a copy of A j for some j ∈ {6, 7}.Now, suppose j = 7.Let a, b, c and d be the vertices of A 7 , as they are labelled in Figure 5.2.There are two edges connecting b to a. Without loss of generality, we may assume the darts on one of these edges have trivial voltage, as necessarily one of these edges must lie on a spanning tree of ∆ and (we can assume) the voltage assignment ζ is simplified.As for the other edge, let r = 0 be the voltage of the dart underlying it and beginning at b (and thus, its inverse, beginning at a, has voltage −r).Let ι(c) = m so that ι(b) = ι(a) = 2m.Observe that (c 0 , b 0 , a 0 , a m , b m ) and (c 0 , b 0 , a r , a m+r , b m ) are 5-cycles of Γ (note that this is true even if r = m).Then every dart beginning at b 0 lies on a 5-cycle.Since dc is a [1, i]-edge of (∆, λ), we see that d 0 is adjacent to c 0 in Γ.Furthermore, since Γ is vertex-transitive, the dart beginning at d 0 and ending at c 0 lies on a 5-cycle C := (d 0 , c 0 , u, v, w), for some u, v, w ∈ V(Γ).Clearly, u ∈ {b 0 , b m } and v ∈ {a 0 , a r , a m , a m+r }, since both fib(c) and fib(b) are independent sets.Then w ∈ fib(a) ∪ fib(b) which leads us to a contradiction, since no vertex in fib(a) ∪ fib(b) is adjacent to a vertex in fib(d).Therefore, there is no 5-cycle tracing the edge d 0 c 0 and thus Γ is not vertex-transitive.
Finally, suppose j = 6 and assume the notation in Figure 5.2.Then ι(d) = m for some m ∈ N. By Lemma 5.6, Γ contains a 3-cycle and thus the vertex d 0 ∈ fib(d) must lie on a 3-cycle C. Since d 0 has two neighbours in fib(c), one vertex of C must be in fib(c).Without loss of generality, let c 0 be that vertex.Then, the third vertex in C must be a common neighbour of d 0 and c 0 , but the other two neighbours of c 0 are b 0 and b 2m , none of which si adjacent to d 0 .Therefore, d 0 does not lie on a 3-cycle, and Γ is not vertex-transitive.

The set Q
Recall that G is the set of all vertex-transitive cubic graphs Γ of order n > 20 admitting an automorphism g of order n/3 or greater, and that Q is the set of all labelled quotients Γ/ g .As a step towards proving Theorem 1.1, we must determine the set Q.Then, we can reconstruct G by considering all vertex-transitive ccv-covers of elements of Q. Observe that Q is a subset of the set Q * of all labelled graphs satisfying the conditions stated in Theorems 4.2 and 5.9.These conditions are restrictive enough to allow us to quickly compute Q * by means of a brute-force algorithm.As it transpires, Q * consist of 20 labelled graphs.To determine Q it suffices to determine which of these graphs admit a vertex-transitive ccv-cover with more than 20 vertices.
The eight elements of Q * having less than four vertices, shown in Figure 6.1, correspond to the eight possible quotients of a cubic graph by a (k, )-semiregular automorphism with k ∈ {1, 2, 3} (graphs with such an automorphism are also called k-multicirculants).
Each of these eight graphs admit at least one vertex-transitive ccv-cover with more than 20 vertices (see [23,Theorem 1.1] or [20] and [22] for details).Therefore, the eight graphs of Figure 6.1 are elements of Q.The remaining 12 labelled graphs of Q * are shown in Figure 6.2.We will show that, with the exception of ∆ 12 , none of these graphs admit a vertex-transitive ccv-cover of order larger than 20.The graph ∆ 12 will be studied in detail in Section 6.1.Now consider a labelled graph ∆ i from Figure 6.2 and suppose (∆, λ, ι, ζ) is a ccv-extension of ∆ i .We may assume that ζ is a simplified voltage assignment and agrees with Figure 6.2, where we adopt the following notation convention.For a symbol α ∈ {r, s}, an edge with an arrow oriented from, say, u to v, with the letter α next to it, indicates that the dart x underlying this edge and beginning at u has voltage α, for some 0 ≤ α < ι(v).A loop with the letter α next to it, indicates that one of the underlying darts, say x, has voltage α for some integer 0 < α < ι(beg x).For a semi-edge x, ζ(x) = ι(beg x)/2.All other darts belong to a spanning T and have trivial voltage.The vertices of each ∆ i are named in Figure 6.2, but we refrain from naming the darts in the figure so as not to overburden it.Since parallel darts in a ccv-graph need to have distinct voltages, every dart in ∆ i is completely determined by its endpoints along with its voltage.Hence, we will denote a dart x of ∆ i beginning at u and ending at v by (uv) ζ(x) ; its inverse is then (vu) −ζ(x) .As every cover of a ccv-graph is a simple graph, the darts in the fibre of (uv) ζ(x) are denoted by u i v i+ζ for i ∈ {0, . . ., ι(u) − 1} (like arc or directed edges are usually denoted).Let (∆, λ) be a labelled graph, Γ be a ccv-cover of (∆, λ) and let π : Γ → ∆ be the corresponding projection.If W is a uv-walk in ∆, then a lift of W based at a vertex u i ∈ fib(u) is a walk W = (x 1 , x 2 , . . ., x n ) beginning at u i such that the projection π(W ) := (π(x 1 ), π(x 2 ), . . ., π(x n )) is equal to W .We denote by L(W ) the set of all lifts of W based at u 0 .
Let (∆, λ, ι, ζ) be a ccv-graph and set n = lcm{λ(x)ι(beg x) : x ∈ D(∆)}.Let W = (x 1 , x 2 , . . ., x k ) be a walk in ∆ and let d = gcd{ι(beg x i ) : x i ∈ W }. We define the endset of W as where d denotes the subgroup of Z generated by d, and where the addition is computed modulo ι(end x k ).Lemma 6.1.[24, Lemma 30] Let (∆, λ, ι, ζ) be a ccv-graph and Γ = Cov(∆, λ, ι, ζ).Let W be a uv-walk for some u, v ∈ V(∆).If W ∈ L(W ), then the final vertex of W is v j for some j ∈ end(W ).Conversely, for every j ∈ end(W ) there exists a lift of W beginning at u 0 and ending at v j .Lemma 6.2.Let (∆, λ, ι, ζ) be a ccv-graph and Proof.Let C be a cycle in Γ.Clearly, C is a u a u a -walk for some vertex u a ∈ fib(u) and some u ∈ V(∆).Let x be a dart visited by π(C) and suppose that π(C) traces x −1 immediately after x.We will show that λ(x −1 ) = 1.Observe that since π(C) traces x and x −1 one after the other, there must exist a dart x i ∈ fib(x) and a dart (x −1 ) j ∈ fib(x −1 ) such that C traces x i and (x −1 ) j consecutively.Since C is a cycle, it is a reduced walk by definition, and thus (x −1 ) j = (x i ) −1 .However both (x −1 ) j and (x i ) −1 belong to fib(x −1 ).Moreover, since x i and (x −1 ) j are two consecutive darts of a walk, we have beg(x −1 ) j = end x i = beg(x i ) −1 .That is, there are two distinct darts in fib(x −1 ) beginning at the same vertex.This implies that λ(x −1 ) ≥ 2. Now suppose x and x −1 are the first and the last darts traced by π(C).Let x i ∈ fib(x) and (x −1 ) j ∈ fib(x −1 ) be the first and last darts traced by C, respectively.By an argument analogous to the one used in the previous case, we have that (x −1 ) j = x −1 i but beg(x −1 ) j = beg x −1 i , and thus λ(x) ≥ 2. This shows that π(C) is λ-reduced.
To show that 0 ∈ end(π(C)), recall that Γ admits an automorphism ρ that maps every dart x i to x i+1 .Then ρ −a maps the vertex u a to u 0 , and thus ρ −a (C) is a reduced closed walk beginning and ending at u 0 .Moreover, ρ −a (C) ∈ L(π(C)).Since the final vertex of ρ −a (C) is u 0 , it follows from Lemma 6.1 that 0 ∈ end(π(W )).
Throughout the rest of the section we will assume that (∆, λ) = ∆ i for some i ∈ {1, . . ., 11}, that Γ is the cover of a ccv-extension (∆, λ, ι, ζ) of ∆ i , and that π : Γ → ∆ is the covering projection.Note that Γ is completely determined by the values of the voltages r and s, and m := gcd{ι(u) : u ∈ V(∆)}.Indeed, Γ is uniquely determined by the quadruple (∆, λ, ι, ζ).Recall that the index function ι is determined by its value on a single vertex along with the labelling λ, which is given.Let u ∈ V(∆) be any vertex and for each v ∈ V(∆) \ {u} chose (arbitrarily) a uv-walk W v .By equality (4.3), we have Note that c depends only on λ and our choice of u.Then, ι is completely determined by λ and m.Finally, since we can assume ζ to be simplified, we know that every dart x underlying a semi-edge has voltage ι(beg x)/2, and any other dart not labelled r or s has voltage 0. The values of r and s, along with the function ι thus completely determine ζ.
We are now ready to analyse the labelled graphs ∆ i .The technique employed in the following pages relies mainly in finding a closed walk W of length n in ∆ i such that L(W ) contains a cycle of length n, regardless of the specific values of the voltages r and s.Such a walk can often be found by finding an artefact A j in ∆ i .If we suppose that Γ is vertex-transitive, then for every vertex v i of Γ, at least one dart incident to v i must lie on an n-cycle.Then by Corollary 6.2, this will imply that for every vertex v ∈ V(∆) a specific dart incident to v lies on a closed walk W of length n such that 0 ∈ end(W ).Since every element of end(W ) can be seen as a linear combination of m, r and s, 0 ∈ end(W ) implies a relation between m, r and s, which along with the fact that gcd(m, r, s) = 1 (see item (5) of Lemma 3.8), is often enough to completely determine their values (up to a few options).This, in turn, determines the graph Γ. Lemma 6.3.If Γ is a vertex-transitive ccv-cover of ∆ 1 , then Γ is a bicirculant graph of order 12.
For each of the two cases in the proof of Lemma 6.5, a subgraph of Γ containing 4 distinct 6-cycles through the dart d 0 a 0 (left) and through the dart e 0 c 0 (right), respectively.
Finally, suppose suppose j = 11.Let ι(d) = m and note that since the voltage assignment is simplified, the darts incident to a or b have voltage m and those incident to c or d have voltage m/2.Since Γ is connected, gcd(m/2, m) = 1, which implies that m = 2.Note that the functions ζ and ι are thus completely determined, and so is the graph Γ.One can simply verify that Γ is a non-vertex-transitive graph of order 12. Lemma 6.5.If Γ is a vertex-transitive ccv-cover of ∆ j , with j ∈ {4, 5}, then Γ has less than 20 vertices.
Observe that every dart in e 0 c 0 lies on 4 distinct 6-cycle in L(W ) (see Figure 6.3, right) and by Lemma 5.4, Γ has less than 20 vertices.Lemma 6.6.If Γ is a vertex-transitive ccv-cover of ∆ 6 , then Γ is a tricirculant of order 18.
Let (α, β, γ), α ≤ β ≤ γ, be the 6-signature of Γ.Note that the edge {u 0 , a 0 } lies on both of the cycles C 1 and C 2 , while each of the remaining two edges incident with u 0 lies on precisely one of them.This shows that α, β ≥ 1 and γ ≥ 2. Similarly, since both C 1 and C 2 pass through the edges {a 0 , u 0 } and a 0 , u m }, incident with a 0 , we see that β, γ ≥ 2.Moreover, since α ≥ 1, there must a third 6-cycle C 3 passing through the edge {a 0 , b s }.Since C 3 also passes through one of the edges {a 0 , u 0 }, {a 0 , u m }, it follows that γ ≥ 3. Finally, since Γ admits the automorphism ρ m (where ρ is the canonical covering transformation) of order 2 fixing a vertex and swapping two of its neighbours, we see that two of the parameters α, β, γ must be equal.By Lemma 5.1 (see also [27,Theorem 1]), it follows that Γ ∼ = SDW(m, 3).Finally, as mentioned in Remark 5.2 (and proved in [27, Proposition 5]), the automorphism group of SDW(m, 3), m = 3, equals S 3 × D m and thus contains an element of order 2m if and only if m is odd.
To conclude the proof of the theorem, assume that m is odd, m > 3, and consider the mapping ϕ : V(Γ 12 (m, 1, 2)) → V(SDW(m, 3)) given by for all i ∈ {0, 1, . . ., m − 1}, where the indices at b j 's and a j 's are computed modulo m, while those at u j 's and v j 's are computed modulo 2m.The isomorphism ϕ is depicted in Figure 6.7 where each vertex of SDW(m, 3) is labeled with its ϕ-image.It is obvious that ϕ is a graph isomorphism.Using the results proved in the previous sections, it is now not difficult to prove Theorem 1.1.Before we proceed to the proof, we will need the lemma below, characterizing those cyclic Haar graphs that are circulant graphs.Proof.Suppose that Λ = H(m, x, y), gcd(m, x, y) = 1, is a circulant.For i ∈ Z m , we let u i and v i denote the vertices (i, 0) and (i, 1), respectively, and recall that the neighbours of u i in Λ are v i , v i+x and v i+y .
Since Λ is a circulant, it is isomorphic to a prism or to a Möbius ladder.In both cases, the girth of Γ is 4 and at every vertex there is an edge belonging to two 4-cycles, none of these 4-cycles sharing a 2-path (recall that we are assuming that m ≥ 5 and thus the order of Λ is at least 10).Observe also that exactly one of the graphs Λ, H(m, −x, y − x) and H(m, x − y, −y), isomorphic to Λ, is such that the edge incident with the vertex (0, 0) belonging to two 4-cycles is e 0 = {u 0 , v 0 }.Denote this graph H(m, a, b) (that is, (a, b) is one of (x, y), (−x, y − x) or (x − y, −y), or equivalently, (x, y) is one of (a, b), (−a, b − a) or (a − b, −b)).
One of the two 4-cycles passing through e 0 passes also through the edge e 1 = {u 0 , v a }, while the other passes through e 2 = {u 0 , v b }.Since the neighbours of v 0 are u 0 , u −a and u −b , for these two edges to form a 4-cycle together with the edge e 0 , either u −a is adjacent to v a and u −b to v b , or u −b is adjacent to v a and u −a to v b .In the first case, both 2a and 2b belong to the set {0, a, b}, while in the second case, a + b ∈ {0, a, b}, which forces a + b = 0 (since a = 0 = b).
Let us consider the first case.Observe that 2a = a and 2b = b.If 2a = 0, then m is even, a = m/2 and 2b = a.Since gcd(m, a, b) = 1, this forces m = 4, contradicting our assumption that Λ has at least 10 vertices.A similar contradiction is obtained if 2b = 0.This leaves us with the possibility that 2a = b and 2b = a.But then a + b = 0, and we find ourselves in the second case.
To summarise, if a connected cubic cyclic Haar graph H(m, x, y) with m ≥ 5 is a circulant then {x, y} = {a, 2a} or {x, y} = {a, −a} for some a ∈ Z m with gcd(m, a) = 1.Furthermore, the graphs H(m, a, 2a) and H(m, a, −a) are isomorphic to the prism Prism(m) whenever m is even, and thus cannot be circulants.It follows that m must be odd, as required.
For the converse, let m, a ∈ Z be such that m ≥ 5, m is odd and gcd(m, a) = 1.If Λ = H(m, a, −a) then the mapping given by u i → v i+r and v i → u i+r is a circulant automorphism of Λ.On the other hand if Λ = H(m, a, 2a) then the mapping given by u i → v i and v i → u i−2r is a circulant automorphism of Λ.
We are now ready to prove Theorem 1.1.As mentioned in Section 2.3, the validity of the theorem for graphs on at most 20 vertices can easily be checked by consulting the census of cubic vertex-transitive graphs [21].Now, as in the statement of Theorem 1.1, let Γ be a cubic vertex-transitive graph of order n, n > 20, admitting an automorphism of order at least n 3 .We need to show that then one of the claims (1) -(4) of Theorem 1.1 holds.
If Γ is a prism or a Möbius ladder, then it is not a circulant only if it is isomorphic to a prism Prism(m) with m even (and then claim (2a) of Theorem 1.1 holds).Further, since the girth of a generalised Petersen graph GP(m, r) is at least 5 unless m ≤ 4 or r = 1, we see that a generalised Petersen graph GP(m, r) is a circulant if and only if r = 1 and m is odd (in which case it is a prism).In particular, if Γ is a generalised Petersen graph which is not a circulant, then claim (2b) of Theorem 1.1 holds.
If Γ is a connected cyclic Haar graph, that is Γ = H(m, r, s) with m > 10 and gcd(m, r, s) = 1, then by applying the permutation ϕ α,0 for an appropriate α, we may assume that r divides m (where r is represented as a positive integer smaller than m), and thus that gcd(r, s) = 1.Under this assumption, by Lemma 7.1 Γ is then a circulant if and only if m is odd and {r, s} = {1, 2} or {r, s} = (1, m − 1).We have thus shown that the claim (2c) of Theorem 1.1 holds in this case.
If Γ is a tricirculant but not a bicirculant, then by [22, Theorems 1.1, 4.3 and 5.3], Γ is either the Tutte's 8-cage (on 30 vertices), the truncated tetrahedron (on 12 vertices) or isomorphic to one of the graphs X(m) or Y(m) with m ≡ 3 (mod 6).Note that claims (3) of Theorem 1.1 holds in this case.
For the converse, it is clear that the circulants, bicirculants and tricirculant appearing in parts (1), ( 2) and (3) of Theorem 1.1 all admit an automorphism of order at least one third of the order of the graph.The graph SDW(m, 3) with m ≡ 3 (mod 6) has order 6m and admits, by construction, an automorphism with two orbits of size 2m, namely, the canonical covering transformation ρ.This completes the proof of Theorem 1.1.

Relationship between κ(Γ) and η(Γ) and open problems
Let us conclude the paper with a discussion on the interplay between the parameters κ(Γ) and η(Γ).First, note that none of the functions η(Γ) and κ(Γ) can be bounded above by a constant, even when restricted to the class of cubic vertex-transitive graphs.Namely, if there were a constant C such that η(Γ) ≤ C holds for all graphs Γ, then max{o(g) : g ∈ Aut(Γ)} ≥ |V(Γ)|/C, or in other words, the parameter meo(Γ) := max{o(g) : g ∈ Aut(Γ)} is bounded below by a linear function of |V(Γ)|.However, it was shown in [25] that there exists an infinite family {Γ i } i∈N of cubic vertex-transitive graphs with meo(Γ i ) growing slower than any logarithmic function of |V(Γ i )|.This implies that there is no constant C such that η(Γ) ≤ C for all cubic vertex-transitive graph Γ.Since η(Γ) ≤ κ(Γ) holds for all Γ, this implies that there is no upper bound on κ(Γ).
Let us mention at this point that a somewhat similar problem was raised in [4], where it was conjectured that the value m(n) := min{ |V(Γ)| κ(Γ) : Γ a cubic vertex-transitive graph on at most n vertices} tends to ∞ as n grows to ∞, or in other words, for every integer n, there is a constant c n such that every cubic vertex-transitive graph on more than n vertices admits a semiregular element of order at least c n .However, in [28], an infinite family of cubic vertex-transitive graphs was constructed in which every semiregular automorphism has order at most 6, proving this conjecture wrong in general (but proved to be correct when restricted to arc-transitive or Cayley graphs; see also [18]).While κ(Γ) can take an arbitrary large value, a question arises whether the parameter κ(Γ) can be bounded above in terms of η(Γ) where Γ ranges through the class of cubic vertex-transitive graphs.More precisely, we are interested in the following: Question 8.1.For which positive integers r does there exist an integer k r such that κ(Γ) ≤ k r for all but finitely many cubic vertex-transitive graphs satisfying η(Γ) ≤ r.
If for some r an integer k r as above exists, then let f (r) be the smallest such integer k r .A direct consequence of Theorem 1.1 is that f (r) can be defined at least for r ∈ {1, 2, 3} and that f (1) = 1, f (2) = 2 and f (3) = 6.Let us conclude this paper by posing the following:

Figure 2 . 2 .
Figure 2.2.The nine labelled graphs in Q, where darts with no label have λ-value 1.With the exception of the right-most graph, these graphs correspond to the 8 possible quotients of a k-multicirculant graph by a k-multicirculant automorphism, with k ∈ {1, 2, 3}.

Lemma 4 . 1 .
Let Γ be a cubic vertex transitive graph of order n > 20, and let G ≤ Aut(Γ) be a cyclic group with an orbit of size n/3 or greater.Then G has at most 5 orbits on vertices.Now, consider the function λ * defined with formula(3.4).If W = (x 1 , x 2 , . . ., x n ) is a uv-walk in ∆, then by a consecutive application of equality (3.1) to the darts of W we have

Figure 5 . 2 .
Figure 5.2.The two artefacts A 6 and A 7 (bottom row).Above each, a small subgraph of the cover of any ccv-graph containing it.

Figure 6 . 4 .
Figure 6.4.The 8 distinct 8-cycles through the dart d 0 a 0 in the proof of Lemma 6.10

Figure 6 . 5 .
Figure 6.5.Two subgraphs of Γ in the proof of Lemma 6.10.On the left, a subgraph containing all lifts of W 1 that visit c 0 (8-cycles through c 0 b 0 shown in bold edges).On the right, a subgraph containing all lifts of W 3 and W 4 that visit c 0 .

Lemma 7 . 1 .
For an integer m ≥ 5, a connected cyclic Haar graph H(m, x, y) is a circulant if and only if m is odd and {x, y} = {a, 2a} or {x, y} = {a, −a} for some a ∈ Z such that gcd(m, a) = 1.