Consistent Cycles in Graphs and Digraphs

Consistent cycles in graphs and digraphs

Stefkoˇ Miklaviˇc Department of Mathematics and Computer Science Faculty of Education University of Primorska Slovenia PrimoˇzPotoˇcnik Department of Mathematics Faculty of Mathematics and Physics University of Ljubljana Slovenia Steve Wilson Department of Mathematics and Statistics Northern Arizona University USA

February 17, 2006

Abstract Let Γ be a ﬁnite digraph and let G be a subgroup of the automorphism group of Γ. A directed cycle C~ of Γ is called G-consistent whenever there is an element of G whose restriction to C~ is the 1-step rotation of C~ . Consistent cycles in ﬁnite arc-transitive graphs were introduced by Conway in one of his public lectures. He observed that the number of G-orbits of G-consistent cycles of an arc-transitive group G is precisely one less than the valency of the graph. In this paper, we give a detailed proof of this result in a more general settings of arbitrary groups of automorphisms of graphs and digraphs.

1 Introduction

Let Γ be a ﬁnite digraph (without loops and multiple arcs). By a directed cycle in Γ we mean a connected subdigraph of Γ where every vertex has in- and out-degree equal to 1. Let G be a subgroup of the automorphism group of Γ. Clearly, G acts on the set of directed cycles. The orbits of this action will be called G-congruence classes of directed cycles, and two directed cycles will be called G-congruent if they belong to the same G-orbit. The cycles of length 2 will be called trivial. By a simple closed walk of Γ we mean a sequence (w0, w1, . . . , wr−1, w0)(r ≥ 2) of vertices of Γ, such that (wi, wi+1)(i ∈ Zr) are arcs of Γ and wi 6= wj for distinct elements i, j ∈ Zr. Let α = (w0, w1, . . . , wr−1, w0) be a simple closed walk of Γ. Then α is G-consistent, if there g exists g ∈ G, such that wi = wi+1 for every i ∈ Zr.

1 ~ If C is a directed cycle with vertices {w0, . . . , wr−1} and arcs (wi, wi+1)(i ∈ Zr), then (w0, . . . , wr−1, w0) is a simple closed walk, called a representative of C~ . A directed cycle C~ will be called G-consistent whenever one (and thus each) of its representatives is G-consistent. Consistent cycles in finite arc-transitive graphs were introduced by Conway in one of his public lectures. He also presented a surprising result, which states that for any finite graph Γ admitting an arc-transitive group of automorphisms G, the number of G-congruence classes of non-trivial G-consistent directed cycles is k − 1, where k is the valency of Γ. To the best of our knowledge, the only written record of this result can be found in Biggs’ paper [1]. A sketch of the proof given there is based on a beautiful idea of Conway employing a recursive construction of a rooted tree, the leaves of which correspond to the G-congruence classes of the non-trivial (directed) consistent cycles, and a clever choice of a flow on the tree, whose only source (with capacity k −1) is the root, and the sinks (each of capacity 1) are the leaves of the tree. By taking a slightly different approach (but still using a similar trick), we generalize the above result in two ways. Firstly, we will assume that Γ is a digraph (rather than a graph), and secondly, we will allow G to be an arbitrary group of automorphisms of Γ. In this setting, we prove that, for any vertex v of Γ, the number of G-congruence classes of G-consistent closed walks starting (and ending in v) equals |vG ∩ Γ+(v)| where Γ+(v) is the out-neighbourhood of v, and vG is the G-orbit of v. (See Corollary 3.5 for a precise statement.) In particular, if G acts transtively on vertices of Γ, then the number of G- congruence classes of G-consistent directed cycles equals the out-valency of the digraph. (see Theorem 4.1.) Many results in this paper follow Biggs [1], but for the sake of self- containedness, the detailed proofs are presented here. Our paper is organized as follows: in the next section we review some definitions and basic notations concerning digraphs. In Section 3 we introduce consistent walks and derive some of their properties. In Section 4 we prove our main result. Finally, in the last section a few examples are given.

2 Preliminaries

In this section we review some definitions and basic notations concerning digraphs. A (finite) digraph is a pair (V,A) where V is a finite set of vertices and A ⊆ V × V \{(v, v) | v ∈ V } is a set of its arcs (or also darts). If Γ is a digraph, then V Γ and AΓ denote its vertex-set and its arc-set, respectively. An arc (u, v) ∈ AΓ is said to be undirected if (v, u) ∈ AΓ; in this case we also say that {u, v} is an edge of Γ. A graph is a digraph with all the arcs undirected. The set of edges {{u, v} | (u, v) ∈ AΓ} of a graph Γ will be denoted by EΓ. For the rest of this section let Γ be a digraph. A walk of length r (or an r-walk) from u to v in Γ is a sequence (w0, . . . , wr) of vertices wi ∈ V Γ such that u = w0, v = wr, and (wi−1, wi) ∈ AΓ, for 1 ≤ i ≤ r. Note that for every vertex v in Γ, the sequence (v) is a walk in Γ of length 0. A walk (w0, w1, . . . , wr) of length r ≥ 2 is closed if wr = w0, and is simple if wi 6= wj for all i, j, 0 ≤ i < j ≤ r, except possibly for (i, j) = (0, r). Closed walks of length 2 are called trivial. For a walk α = (w0, . . . , wr) and for an integer i ∈ {0, . . . , r} we denote α[i] = wi. Moreover, if r ≥ 1, then we let αb = (w0, . . . , wr−1). For a vertex v ∈ V Γ we define the out- and in-neighbourhood of v by Γ+(v) = {u ∈ V Γ | (v, u) ∈ AΓ} and Γ−(v) = {u ∈ V Γ | (u, v) ∈ AΓ}, respectively. We let δ+(v) = |Γ+(v)| and

2 δ−(v) = |Γ−(v)| denote the out- and in-degree of v. A rooted tree is a digraph T with exactly one vertex w of in-degree 0 (called the root of T ), and with the property that for every vertex v ∈ VT there exists exactly one walk from w to v. (Note that the underlying graph of a rooted tree is a tree, and conversely, every tree with a designated root determines a rooted tree in a natural way.) Leaves of a rooted tree are vertices with out-degree 0. Note that every (ﬁnite) rooted tree has at least one leaf. An internal vertex of a rooted tree is a vertex which is not a leaf. Let g be a permutation of V Γ. For a vertex v ∈ V Γ let vg denote the image of v under g. An automorphism of Γ is a permutation g of V Γ, such that (u, v) ∈ AΓ if and only if (ug, vg) ∈ AΓ. The group of all automorphisms of Γ is called the automorphism group of Γ and will be denoted by Aut(Γ). If G is a subgroup of Aut(Γ), then we say that Γ is G- vertex-transitive (respectively, G-arc-transitive), if G acts transitively on V Γ (respectively, AΓ). Furthermore, if Γ is a graph and G acts transitively on EΓ, then we say that Γ is G-edge-transitive. The symbol G will be omitted from this notation if G = Aut(Γ). Finally, for g, h ∈ G, we let gh = h−1gh denote the h-conjugate of g. For v ∈ V Γ and G ≤ Aut(Γ), we let vG = {vg | g ∈ G} denote the G-orbit of v, and g Gv = {g ∈ G | v = v} the stabilizer of v in G. Furthermore, for a walk α = (w0, . . . , wr) in

Γ we let Gα = Gw0 ∩ Gw1 ∩ · · · ∩ Gwr .

3 Consistent walks

In this section we introduce the notion of consistent walks and consistent cycles in digraphs and derive some results about them. Let Γ be a digraph and G ≤ Aut(Γ). A walk α = (w0, . . . , wr) in Γ is called G-consistent (or just consistent if the subgroup G is clear from the context) if there exists g ∈ G such g that wi = wi+1 for i ∈ {0, 1, . . . , r − 1}. In this case we say that g is a shunt automorphism for α. The set of all shunt automorphisms g ∈ G for α will be denoted by ShG(α). It is easy to see that consistent walks are essentially simple in the following sense: If α = (w0, . . . , wr) is a G-consistent walk in Γ such that wi1 = wi2 for some integers i1 < i2, then for some positive integer s we have that (w0, . . . , ws) is a simple closed walk and wi = wj whenever i ≡ j mod s. For the sake of simplicity, we shall abuse the notation slightly and use the term G-consistent closed walk only for simple closed walks.

Lemma 3.1 Let Γ be a digraph, let G ≤ Aut(Γ) and let α = (w0, . . . , wr) be a G-consistent walk in Γ. Then ShG(α) is closed under conjugation by the elements of Gα.

Proof. Let g be a shunt automorphism for α, and let h be an element of Gα. Then for h−1gh gh h h every i ∈ {0, . . . , r − 1} we have that vi = vi = vi+1 = vi+1. Therefore, g is also a shunt automorphism contained in G, and the result follows.

For a digraph Γ, for a subgroup G ≤ Aut(Γ), and for a G-consistent walk α = (w0, . . . , wr), we let Gαb RG(α) = wr . (1)

Now suppose that g1, g2 ∈ G are two distinct shunts for α, and for i ∈ {1, 2} consider the i gi 2 sets XG(α) = {u | u ∈ RG(α)}. If v ∈ XG(α), then there exists an element h ∈ Gαb such

3 −1 −1 hg2 hg2 hg2g1 g1 g hg2g1 that v = wr . But then v = wr = wr = u 1 , where u = wr . However, since −1 1 2 1 hg2g1 ∈ Gαb, it follows that u ∈ RG(α). Thus v ∈ XG and so XG ⊆ XG. We can use the 1 2 2 1 same argument to show XG ⊆ XG, and thus XG = XG. In particular, the set g XG(α) = {u | u ∈ RG(α)} (2) is independent of the choice of the shunt g ∈ ShG(α), and therefore hg XG(α) = {wr | h ∈ Gαb, g ∈ ShG(α)}. (3)

Lemma 3.2 Let Γ be a digraph, let G ≤ Aut(Γ) and let α = (w0, . . . , wr) be a G-consistent walk of Γ. Then the set XG(α) is invariant for the action of the stabilizer Gα.

Proof. Let v ∈ XG(α) and f ∈ Gα. Then, by (3), there exist elements h ∈ Gαb and hg g ∈ ShG(α) such that v = wr , and hence

f hgf hfgf v = wr = wr . f f But since hf ∈ Gαb and g ∈ ShG(α) by Lemma 3.1, (3) implies that v ∈ XG(α), as asserted.

Let α = (w0, . . . , wr) be a G-consistent walk of Γ and let w be a vertex of Γ. If β = (w0, . . . , wr, w) is a G-consistent walk of Γ then it is called a G-consistent extension of α. We have the following lemma.

Lemma 3.3 Let Γ be a digraph, let G ≤ Aut(Γ) and let α = (w0, . . . , wr) be a G-consistent walk of Γ. Then w ∈ XG(α) if and only if (w0, . . . , wr, w) is G-consistent. In particular, the number of G-consistent extensions of α is |XG(α)| = |RG(α)| = |Gαb|/|Gα|. hg Proof. If w ∈ XG(α), then w = wr for some h ∈ Gαb and some shunt automorphism g for α. But then hg is a shunt automorphism for (w0, . . . , wr, w). On the other hand, suppose 0 g0 that (w0, . . . , wr, w) is G-consistent walk with a shunt automorphism g . Then w = wr and since wr ∈ RG(α), we have w ∈ XG(α).

3.1 The tree of consistent cycles For a digraph Γ and G ≤ Aut(Γ) let H be a subgroup of Aut(Γ) normalizing G, and let α = (w0, . . . , wr) be a G-consistent walk in Γ with a shunt automorphism g. If h ∈ H, then h h h h α = (w0 , . . . , wr ) is clearly a G-consistent walk with a shunt automorphism g . Hence the group H partitions the set of G-consistent walks in Γ into H-congruence classes. In this subsection, we modify Conway’s approach slightly and prepare the stage for our main result. The idea behind Conway’s approach is to define a rooted tree which encodes all the informations about the structure of congruence classes of consistent cycles in a given digraph and for a given group of automorphisms. For a rooted tree T with the root ω and ν ∈ VT , let αν be the unique walk from ω to ν. Furthermore, for a function ι from VT to an arbitrary set V , and for a walk α = (ν0, . . . , νr) in T , let ι(α) = (ι(ν0), . . . , ι(νr)). In what follows, we fix a digraph Γ, a group G ≤ Aut(Γ), and a vertex v0 ∈ V Γ. We define a rooted tree T (with the root denoted by ω) and functions ι: VT → V Γ, `: AT → N with the following properties:

4 (T1) ι(αν) is a G-consistent walk in Γ for every ν ∈ VT ;

(T2) if ι(αν) = (v0, . . . , vr) and vi = vj for some i, j (0 ≤ i < j ≤ r), and ν ∈ VT , then i = 0, j = r, and ν is a leaf of T ;

(T3) `(η, ν) = |RG(ι(αν))| for every (η, ν) ∈ AT ; P (T4) ξ∈T +(ν) `(ν, ξ) = |XG(ι(αν))| for every internal vertex ν ∈ VT . We shall deﬁne the rooted tree T and functions ι and ` recursively by deﬁning triples (Tr, ιr, `r), 0 ≤ r ≤ s, where Tr is a rooted tree with the root ω, and ιr : VTr → V Γ, `r : ATr → N are such that:

(a) Tr ≤ Tr+1;

(b) ιr and `r are restrictions of ιr+1 and `r+1 on VTr and ATr, respectively;

Because of (b) above, we shall abuse the notation slightly and write ι and ` instead of ιr and `r (0 ≤ r ≤ s). Let T0 be the rooted tree with a single vertex ω, labeled by ι(ω) = v0, and no arcs. Observe that the conditions (T1)–(T4) are trivially satisfied in this case. Suppose now that r ≥ 0 and that the triple (Tr, ι, `) is already defined in such a way that (T1)–(T4) are satisfied. Let Λ = {ω} if r = 0, and let Λ be the set of all the leaves of Tr which are not labeled with v0 if r ≥ 1. If Λ = ∅, then let s = r, T = Ts and quit. Otherwise, define the tree Tr+1 and extend the functions ι and ` to Tr+1 as follows. For every µ ∈ Λ, consider the unique walk αµ = (µ0, . . . , µt) in Tr from µ0 = ω to µt = µ. Since, by the induction hypothesis, ι(αµ) is a G-consistent walk in Γ, the stabilizer Gι(αµ) acts on the set

XG(ι(αµ)) by Lemma 3.2. So we can choose a complete set of representatives {w1, . . . , wmµ } of the Gι(αµ)-orbits on XG(ι(αµ)). For every such representative wi, create a “new” vertex + + + + µi , and a “new” arc (µ, µi ) of Tr+1. Furthermore, define ι(µi ) = wi and let `(µ, µi ) be the size of the Gι(αµ)-orbit containing wi. Note that the condition (T4) clearly holds in Tr+1 for ν = µ an hence for every ν ∈ Λ. On the other hand it is easy to see that, by induction, (T4) holds for all other internal vertices of Tr+1. Similarly, the induction hypothesis implies that (T3) holds in Tr+1 provided it holds for + + every “new” arc (η, ν) = (µ, µi ). Furthermore, by the above construction, `(µ, µi ) is the Gι(αµ) + Gι(αµ) + size of the Gι(αµ)-orbit w , where wi = ι(µ ). Since ι(αµ) = ι\(α ), by (1), w i i µi i + coincides with RG(ι(α + )). Hence `(µ, µ ) = |RG(ι(α + ))| and (T3) holds in Tr+1. µi i µi Finally, let us show that conditions (T1) and (T2) are satisfied for Tr+1. Let ν be a vertex of Tr+1 and let αν = (ν0, . . . , νt). If ν = νt ∈ VTr, then (T1) and (T2) hold by the induction + hypothesis. Therefore we may assume that ν = µi for some µ ∈ Λ and i ∈ {1, . . . , mµ}. Then, by definition of Tr+1, we have wi ∈ XG(ι(αµ)). Hence, by Lemma 3.3, ι(α + ) is µi G-consistent. This shows that (T1) holds in Tr+1. To show that (T2) holds as well, assume that ι(νi) = ι(νj) for some i, j (0 ≤ i < j ≤ t). But then, as we observed in the beginning of this section, i = 0 and j = t. This completes the proof that the triples (Tr, ι, `) satisfy conditions (T1)–(T4).

Observe that (T2) implies that each of the trees Tr has depth at most |V Γ|. Since at each step of the construction, the depth of the tree increases by 1, this implies that the

5 construction terminates after at most |V Γ| steps. If T = Tr is the last tree constructed by this procedure, then the triple (Tr, `r, ιr) is called the tree of consistent cycles of Γ with respect to G and v0. Let us now show that, besides (T1)–(T4), the tree of consistent cycles has some additional nice properties.

Lemma 3.4 Let (T, `, ι) be the tree of consistent cycles of a digraph Γ with respect to G ≤ Aut(Γ) and vertex v. Then the following holds:

(i) For every vertex ν of T , ι(ν) = v if and only if ν = ω or ν is a leaf of T .

(ii) If ν is a leaf of T and η ∈ T −(ν), then `(η, ν) = 1.

(iii) There is a bijective correspondence between the leaves of T and the Gv-congruence classes of G-consistent closed walks in Γ starting (and ending) in v.

Proof. Part (i) follows directly from (T2) and the fact that the procedure of constructing T terminates when all the leaves are labeled by v. To show Part (ii), let ν be a leaf of T and let η be the element of T −(ν). As before, let αν be the unique walk in T from the root ω to the leaf ν, and let α = ι(αν). Since ι(ν) = v, it follows by (T3) and (1) that

Gαb `(η, ν) = |RG(α)| = |v |.

However, since the walk αb starts with v, the stabilizer Gαb fixes v, and thus the right-hand side of the above equality is 1, as claimed in Part (ii) of the lemma. Let us now prove Part (iii). Let L be the set of all the leaves of T , and let C be the set of Gv-congruence classes of G-consistent closed walks in Γ starting in v. We shall prove Part (iii) by defining a pair of functions γ : L → C and δ : C → L, and showing that one is the inverse of the other. The definition of γ is straightforward: For a leaf ν ∈ L, let γ(ν) be the element of C containing ι(αν). (Note that, since ν is a leaf and thus ι(ν) = v, ι(αν) is indeed a closed walk starting in v. It is G-consistent by (T1).) Let us now define the function δ : C → L. Let α = (v0, v1, . . . , vr−1, vr) be a G-consistent closed walk starting and ending in the vertex v = v0 = vr. We shall recursively define a (0) (1) (r) sequence ν0, ν1, . . . , νr of vertices in T , and G-consistent walks α , α , . . . , α , such that for every s ∈ {0, . . . , r} the following two conditions will be satisfied:

(s) (P1) α is a G-consistent walk in Γ which is Gv-congruent to α;

(s) (P2) (ν0, . . . , νs) is a walk in T and ι(νi) = α [i] for every i ∈ {0, . . . , s}.

(0) Let ν0 = ω and α = α. Then, clearly, (P1) and (P2) hold for s = 0. Suppose now that (0) (t−1) for some t ∈ {1, . . . , r}, the vertices ν0, . . . , νt−1 and the walks α , . . . , α are already defined in such a way that (P1) and (P2) are satisfied for each s ∈ {0, . . . , t − 1}. Then ∗ (t−1) let α = (ι(ν0), . . . , ι(νt−1)) be the walk in Γ consisting of the first t vertices of α , and let u = α(t−1)[t] be the (t + 1)-th vertex of α(t−1). Since α(t−1) is G-consistent, so is the ∗ ∗ extension of α by u. Thus, by Lemma 3.3, u ∈ XG(α ). By the construction of T , there exists a unique vertex νt ∈ VT such that u is in the Gα∗ -orbit of ι(νt). Therefore, there

6 g (t) (t−1) g exists g ∈ Gα∗ ≤ Gv such that u = ι(νt). Let α = (α ) . With thus defined νt and α(t), it is now clear that (P1) and (P2) are satisfied also for s = t. By above recursive procedure, we have constructed a walk αT = (ν0, . . . , νr), which is, (r) by (P1) and (P2), such that ι(αT ) = α is G-consistent in Γ and Gv-congruent to α. In particular, since α is closed and starts (and ends) in v, also ι(αT ) is closed and starts (and ends) in v. But then ι(νr) = v, and by Part (i), νr is a leaf of T . We may thus define the Gv δ-image of the Gv-congruence class α to be νr. It remains to show that this definition does not depend on the choice of the representative of αGv . g Suppose therefore that β = α , for some g ∈ Gv. Then β = (w0, . . . , wr−1, wr) for w0 = wr = v and some vertices wi, i ∈ {1, . . . , r − 1}. Let βT = (µ0, . . . , µr−1, µr) be the walk obtained from β in the same way as αT is from α. Then, by definition, δ(β) = µr. If µr = νr, then δ maps the Gv-congruence classes of α and β to the same element, as required. Assume therefore that µr 6= νr, and let t ∈ {0, . . . , r − 1} be the smallest integer such that µt 6= νt. Since µ0 = ω = ν0, it follows that t ≥ 1. Also, since vi = ι(νi) = ι(µi) = wi for all i ∈ {0, . . . , t − 1}, it follows that the automorphism g, mapping α to β, belongs to ∗ the stabilizer Gα∗ of the walk α = (v0, . . . , vt−1). In particular, vt and wt are in the same Gα∗ -orbit. But then it follows directly from the construction of αT and βT that νt = µt, which contradicts our assumption on t. This shows that δ is a well-defined function from C to L. To finish the proof, let us show that the functions γ : L → C and δ : C → L are inverse to each other. Observe first that each leaf ν of T is the δ-image of the Gv-congruence class of ι(αν), where αν is the unique walk from ω to ν in T . Hence δ is surjective, and γ is its inverse Gv provided that it is its left inverse. Now, let α be an element of C, and let αT = (ν0, . . . , νr) Gv be the corresponding walk in T . By definition of δ, we have δ(α ) = νr. Since αT is the Gv Gv unique walk in T from ω to the leaf νr, it follows that γ(δ(α )) = γ(νr) = ι(αT ) . However, Gv Gv Gv Gv by (P1) and (P2), ι(αT ) = α , and so γ(δ(α )) = α . Hence γ is a left inverse of δ, and since δ is surjective, also its inverse. This completes the proof of Part (iii).

Corollary 3.5 For a vertex v of a digraph Γ and a subgroup G ≤ Aut(Γ), the number of Gv-congruence classes of G-consistent closed walks in Γ starting (and ending) in v is |Γ+(v) ∩ vG|.

Proof. Let (T, `, ι) be the consistency tree of Γ with respect to G and v. By (T4), the labeling ` satisﬁes the Kirchoﬀ law at every internal vertex v. Since the only source of ` is the root ω and since the sinks of ` are precisely the leaves of T , it follows that X X X `(ω, ζ) = `(η, ν) ζ∈T +(ω) ν∈Λ η∈T −(ν) where Λ is the set of leaves of T . By Part (ii) of Lemma 3.4, the right-hand side of the above equality equals to |Λ|, and by Part (iii) of the same lemma, to the number of Gv-congruence classes of G-consistent closed walks in Γ starting (and ending) in v. On the other hand, the left-hand side is, by (T4), the number of G-consistent walks of length 1 starting in v. Moreover, a walk (v, u) of length 1 is G-consistent if and only if u ∈ vG. The result follows.

7 4 Congruence classes of directed and undirected cycles

If C~ is a directed cycle of a graph Γ, then the inverse C~ −1 of C~ (that is, the digraph with the same vertices as C~ but with a pair (v, u) being an arc whenever (u, v) is an arc of C~ ) is also a directed cycle of Γ. A pair of inverse directed cycles determines a connected regular subgraph (called a cycle of Γ) in a natural way. The degree of a cycle is 2 if the length (the number of vertices) is greater than 2, and is 1 if the length is 2. The cycles of length 2 will be called trivial. Note that a trivial cycle is both: directed and undirected. Clearly, a group G of automorphisms of a (di)graph Γ acts on the set of the (directed) cycles. The orbits of this action will be called G-congruence classes of (directed) cycles, and two (directed) cycles will be called G-congruent if they belong to the same G-orbit. ~ If C is a directed cycle with vertices {w0, . . . , wr−1} and darts (wi, wi+1), i ∈ Zr, then (w0, . . . , wr−1, w0) is a closed simple walk, called a representative of C~ . Clearly, every closed simple walk is a representative of a unique directed cycle, and every directed cycle of length r has r representatives, each being a cyclic shift of any of them. Similarly, every closed simple walk α = (w0, . . . , wr−1, w0) represents a unique cycle in a natural way, and every non-trivial cycle of length r has 2r representatives, each being a cyclic shift of a chosen representative −1 α = (w0, . . . , wr−1, w0), or a cyclic shift of the inverse α = (w0, wr−1, . . . , w1, w0). If α is a G-consistent closed walk with a shunt automorphism g ∈ G, then its cyclic shift are just images of α by powers of g. Hence, all the representatives of a directed cycle whose one of its representatives is G-consistent are G-congruent (and thus G-consistent). A directed cycle with a G-consistent representative will be called G-consistent. Note that a ~ directed cycle C of length r is G-consistent if and only if the set-wise stabilizer GC~ of its arcs acts on it as a cyclic group of order r. Similarly, if α is a G-consistent representative of an undirected cycle C in a graph Γ, then all the representatives of C are G-consistent (either with respect to the same shunt g as α or with respect to g−1). A cycle C will be called G-consistent if one (and thus each) of its representatives is G-consistent. Note, however, that not all representatives of a G-consistent cycle need to be G-congruent. Namely, it might happen that a representative α is not G- congruent to its inverse α−1 (and thus to any of the cyclic shifts of α−1). A G-consistent cycle in which all of its representatives are G-congruent will be called G-symmetric.A G- consistent cycle which is not G-symmetric will be called G-chiral. Note that a non-trivial cycle C of length r is G-chiral if and only if the set-wise stabilizer of its edges GC acts on C as a cyclic group of order r, and is G-symmetric if and only if GC acts on C as a dihedral group of order 2r. Corollary 3.5 gives an easy method of counting congruence classes of consistent closed walks starting in a given vertex. In this section, we shall count (directed) cycles, rather than walks. The formula for the number of G-congruence classes of G-consistent (directed) cycles is particularly nice when it is assumed that the group G acts transitively on the set of vertices of Γ.

Theorem 4.1 Let Γ be a G-vertex-transitive digraph, and let v be a vertex of Γ. Then the number of G-congruence classes of G-consistent directed cycles of Γ is |Γ+(v)|. Furthermore, if Γ is a graph, and s¯ and c are the numbers of G-congruence classes of G-symmetric and G-chiral cycles in Γ, respectively, then s¯ + 2c = |vG ∩ Γ(v)|.

8 Proof. By Corollary 3.5 and since Γ is G-vertex-transitive, it suffices to show that there is a bijective correspondence between the set Cv of Gv-congruence classes of G-consistent closed walks in Γ starting in v and the set C of G-congruence classes of G-consistent directed cycles in Γ. For a walk α = (v, v1, . . . , vr−1, v) in Γ, let C~ (α) denote the corresponding directed cycle Gv G in Γ. Define the function ϕ: Cv → C by ϕ(α ) = C~ (α) . It is clear that this function is well defined (that is, independent on the choice of the representative of αGv ). Since G is vertex-transitive, every element of C has a representative of the form C~ (α)G where α starts and ends in v. Hence ϕ is surjective. Let us now show that it is also injective. Gv Gv Suppose that ϕ(α ) = ϕ(β ) for two G-consistent closed walks α = (v0, . . . , vr−1, vr) and β = (w0, . . . , wr−1, wr) such that v0 = w0 = vr = wr = v. Then there exists g ∈ G such that αg is some cyclic shift of β. In fact, by multiplying g by some power of a shunt for β, we may assume that g maps α to β. But then g fixes v, and so αGv = βGv . This shows that ϕ is a bijection, and completes the proof of the first part of the theorem. The second assertion of the theorem now follows easily from the observation that every class of G-symmetric cycles in Γ consists of a single G-congruence class of directed cycles, while a class of G-chiral cycles is a union of exactly two G-congruence classes of directed cycles (each containing the inverses of the other). Note that some G-consistent cycles might be trivial (that is, of length 2). Indeed, such a cycle exist whenever there is an automorphism in G swapping two adjacent vertices. For example, such an automorphisms exists whenever G acts transitively on the arcs of Γ. In this case, all the trivial cycles in Γ are G-symmetric, and mutually G-congruent. Also, the value |vG ∩ Γ(v)| in the statement of Theorem 4.1 is then just the valency of Γ. This yields the following interesting result (see also [1]):

Corollary 4.2 Let Γ be a G-arc-transitive graph of valency k, and let s and c denote the numbers of G-congruence classes of non-trivial G-symmetric and G-chiral cycles in Γ, respectively. Then s + 2c = k − 1. In particular, every G-arc-transitive graph Γ with even valency contains a G-symmetric cycle.

5 The Petersen graph

In this section we demonstrate the use of Corollary 3.5, Theorem 4.1 and Corollary 4.2. Let Γ be the Petersen graph. It is well known that the vertex set of Γ can be identiﬁed with the set of all 2-subsets of the set {1, 2, 3, 4, 5}. Two vertices are adjacent if and only if the corresponding 2-subsets are disjoint (see Figure 1). It is easy to see that the symmetric group S5 acts on the vertex set of Γ as an automorphism group in a natural way. In fact, it is well known that the automorphism group of Γ is isomorphic to S5. Therefore, in this section the automorphisms of Γ will be considered as the corresponding elements of S5. This abuse of notation should not couse any ambiguities.

Example 1. Let τ = (1, 2)(3, 5) ∈ S5, and let G = hτi be the subgroup of S5. (Observe that τ represents the reﬂection of Figure 1 about the vertical axes through the vertex {1, 2}.) Since |Γ+(v) ∩ vG| = 0 for v ∈ {{1, 2}, {3, 5}, {3, 4}, {4, 5}, {2, 4}, {1, 4}}, there are no G- consistent closed walks starting and ending in v, by Corollary 3.5. On the other hand, |Γ+(v) ∩ vG| = 1 for v ∈ {{1, 3}, {1, 5}, {2, 3}, {2, 5}}. Therefore, by Corollary 3.5, there is

9 Figure 1: The Petersen graph exactly one G-congruence class of G-consistent closed walks starting and ending in such a v. Since τ is an involution, the G-consistent cycles are all trivial. For example, the only G-consistent closed walk starting and ending in {1, 3} is the trivial walk ({1, 3}, {2, 5}, {1, 3}).

Example 2. Let ρ = (1, 4, 2, 3, 5) ∈ S5, and let G = hρi. (Note that ρ represents a 1-step rotation of Figure 1 about its centre.) By Corollary 3.5, there are exactly two G- congruence classes of G-consistent closed walks starting and ending in v = {1, 2}. The first one is the class of the ”outer closed walk” ({1, 2}, {3, 4}, {2, 5}, {1, 3}, {4, 5}, {1, 2}) (with shunt automorphism ρ). The second one is the G-congruence class of the inverse of the above closed walk (with shunt automorphism ρ−1). Similarly, there are exactly two G-congruence classes of G-consistent closed walks starting and ending in v = {3, 5}. The first one is the G- congruence class of the ”inner closed walk” ({3, 5}, {1, 4}, {2, 3}, {1, 5}, {2, 4}, {3, 5}) (with shunt automorphism ρ2), and the second one is the class of the inverse of the inner closed walk (with shunt automorphism ρ−2). ∼ Example 3. Let G = hρ, τi = D5. Again, by Corollary 3.5, there are exactly two G- congruence classes of G-consistent closed walks starting and ending in v = {1, 2}. The first one is again the class of the outer closed walk ({1, 2}, {3, 4}, {2, 5}, {1, 3}, {4, 5}, {1, 2}). Note that here this closed walk belongs to the same G-congruence class as its inverse, due to the automorphism τ. The second G-congruence class is the trivial closed walk ({1, 2}, {3, 4}, {1, 2}) (with a shunt automorphism τρ). It is left to the reader to verify, that the number of G- congruence classes of G-consistent closed walks starting and ending in w = {3, 5} is also two.

Example 4. Let σ = (1, 5, 2, 3) ∈ S5, and let G = hρ, σi. Note that σ swaps the inner cycle and the outer cycle, while changing the orientation of one of them. The group G ∼ is isomorphic to the affine group Aff(1, 5) = C5 : C4 and acts transitively on the vertices, but intransitively on the edges of Γ. Hence by Theorem 4.1, there are exactly three G- congruence classes of G-consistent directed cycles. The first class is the G-congruence class of the ”outer cycle” {{1, 2}, {3, 4}, {2, 5}, {1, 3}, {4, 5}, {1, 2}}. Note that the ”inner cycle” {{3, 5}, {1, 4}, {2, 3}, {1, 5}, {2, 4}, {3, 5}} is in the same G-congruence class as the outer cycle. The second class is the G-congruence class of the trivial cycle {{1, 2}, {3, 4}, {1, 2}}, while the third one is the G-congruence class of the trivial cycle {{1, 2}, {3, 5}, {1, 2}}. (Ob- serve that these two trivial cycles are not in the same G-congruence class.) ∼ Example 5. Next, let ϕ = (1, 3, 5) ∈ S5, and let G = hρ, τ, ϕi = A5. Observe that G acts transitively on the arcs of Γ. By Corollary 4.2, we have s+2c = 2, where s and c denote

10 the numbers of G-congruence classes of non-trivial G-symmetric and G-chiral cycles in Γ, respectively. Moreover, since τ maps the outer cycle {{1, 2}, {3, 4}, {2, 5}, {1, 3}, {4, 5}, {1, 2}} (which is G-consistent due to the shunt ρ) to the inverse cycle, we have s ≥ 1, and so s = 2 and c = 0. The other G-congrence class of G-symmetric cycles is represented by the cycle {{1, 2}, {3, 4}, {1, 5}, {2, 3}, {4, 5}, {1, 2}}, which is G-consistent due to the shunt automorphism induced by (1, 3, 5, 2, 4) ∈ A5. ∼ Example 6. Finally, let G = Aut(Γ) = S5. By Corollary 4.2, we have s + 2c = 2, where s and c denote the numbers of S5-congruence classes of non-trivial S5-symmetric and S5-chiral cycles in Γ, respectively. Observe that the outer cycle {{1, 2}, {3, 4}, {2, 5}, {1, 3}, {4, 5}, {1, 2}} and the cycle {{1, 2}, {3, 4}, {1, 5}, {2, 3}, {1, 4}, {3, 5}, {1, 2}} are both S5-symmetric, and mutually non-congruent (since their lenghts are distinct). Hence s = 2 and c = 0.

References

[1] N. L. Biggs, Aspects of symmetry in graphs, Colloquia mathematica societatis J`anosBolyai, Szeged (Hungary), 1978, 27 – 35.