Tetravalent arc-transitive locally-Klein graphs with long consistent cycles

European Journal of Combinatorics 36 (2014) 270–281

Contents lists available at ScienceDirect

European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc

Tetravalent arc-transitive locally-Klein graphs with long consistent cycles Primož Potočnik Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, Ljubljana, Slovenia IAM, University of Primorska, Muzejski trg 2, Koper, Slovenia

article

info

Article history: Received 24 February 2012 Accepted 3 July 2013 Available online 22 August 2013

abstract The topic of this paper is connected tetravalent graphs admitting an arc-transitive group of automorphisms G, such that the vertexstabiliser Gv is isomorphic to the Klein 4-group. Such a graph will be called locally-Klein. A cycle in a graph is said to be consistent if there exists an automorphism of the graph that preserves the cycle setwise and acts upon it as a one-step rotation. The main result of the paper is a classification of those locally-Klein graphs that contain a consistent cycle of length more than half the order of the graph. As a side result, we define an interesting family of graphs embedded on the torus or on the Klein bottle, such that the automorphism group of the resulting map has two orbits on the edges, two orbits on the vertices and two orbits on the arcs of the graph. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction All the graphs in this paper are assumed to be finite, simple and connected. A cycle C of a graph

Γ is called consistent provided that there exists an automorphism g of Γ that acts on C as a one-step g rotation, that is, such that vi = vi+1 where C = v0 v1 . . . vm−1 and indices are computed modulo m. The automorphism g is then called a shunt of C and if g belongs to some group G of automorphism of Γ , then C is said to be G-consistent. Consistent cycles were introduced by Conway in one of his public lectures [4] (see also [1]), where a remarkable theorem regarding the number of orbits of consistent cycles in an arc-transitive graph was proved. This concept had been forgotten for several years, but has re-emerged recently (see example [2,8,13,14]) and proved to be useful in the study of graph symmetries (see for example [5,11,10]). In this paper we shall address the following natural, though a bit vague question:

E-mail address: [email protected]. 0195-6698/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ejc.2013.07.001

P. Potočnik / European Journal of Combinatorics 36 (2014) 270–281

271

Question 1.1. What can be said about arc-transitive graphs that contain a long consistent cycle? Of course, the length of a consistent cycle is bounded above by the order of the graph, and the graphs that meet this upper bound are special types of circulant graphs (recall that a circulant Cir(n, S ), where S ⊆ Zn \ {0} and S = −S, is the graph with vertex set Zn and two vertices x, y ∈ Zn adjacent if and only if x − y ∈ S). Since arc-transitive circulants have been classified independently by Kovács [9] and Li [12], Question 1.1 can thus be answered if the adjective ‘‘long’’ is interpreted as ‘‘of length being equal to the order of the graph’’. Besides such trivial observations, it seems difficult to give any general answer to Question 1.1. In this paper, we shall restrict our attention to a special class of connected arc-transitive graphs of valency 4, namely those the automorphism group of which contains a subgroup G acting transitively on the arcs of the graph and with the property that the vertex-stabiliser Gv of an arbitrary vertex v is isomorphic to the Klein group Z22 ; such a pair (Γ , G) will be called an arc-transitive locally-Klein pair. Even though the class of arc-transitive locally-Klein pairs seems very restricted, it in fact plays an important role in several contexts. For example, up to some degenerate exceptions, locally-Klein pairs correspond to the medial graphs of regular maps (we refer the reader to, say, [18] for the theory of regular maps) and studying locally-Klein pairs is thus in some sense equivalent to the very rich theory of regular maps. In this equivalence, faces, Petrie walks and neighbourhoods of vertices of a regular map correspond to consistent cycles in the medial graph. The main result of this paper is a classification of the arc-transitive locally-Klein pairs (Γ , G), such that Γ contains a G-consistent cycle of length more than half the order of Γ . In the course of proving this result, three interesting families of graphs emerge naturally. The first one is the family of wreath graphs W(n, 2), which are in fact the lexicographic products of cycles with edgeless graphs (see [7] for the definition of a lexicographic product of graphs), and can be described as follows. Definition 1.2. Let n and k be integers, n ≥ 3, and let W(n, k) be the graph with vertex-set Zn × Zk and with edges of the form {(x, i), (x + 1, j)} for x ∈ Zn and i, j ∈ Zk .  Note that W(n, k) is tetravalent only when k = 2; we shall therefore restrict our attention to this case only. The second family consists of the so called depleted wreath graphs DW(d, k) (introduced and named in [19]), which appear as exceptions in many different contexts. A depleted wreath graph DW(d, k) is in fact isomorphic to the direct product Cd × Kk (see [7] for the definition of the direct product of graphs) and can thus be described as follows. Definition 1.3. For integers d and k, satisfying d ≥ 3 and k ≥ 2, let DW(d, k) be the graph with vertex-set Zd × Zk and edge-set {{(x, i), (x + 1, j)} : x ∈ Zd , i, j ∈ Zk , i ̸= j}.  Clearly, the graph DW(d, k) is tetravalent only if k = 3; the depleted wreath graphs appearing in this paper will thus always have the parameter k set to 3. The last of the three exceptional families is the family of graphs X (d, s) defined below. Definition 1.4. For positive integers d and s satisfying d ≥ 2 and 1 ≤ s < d, let V be a set of size 3d with elements denoted by u0 , u1 , . . . , ud−1 and v0 , v1 , . . . , v2d−1 , and let X (d, s) be the tetravalent graph on the vertex set V with adjacencies given by:

vj ∼ vj+1 for j ∈ {0, . . . , 2d − 1},

ui ∼ vi , vi+s , vi+d , vi+d+s

and

where indices at vi are computed modulo 2d.

for i ∈ {0, . . . , d − 1},



As we shall show in Section 2, the families of depleted wreath graphs and graphs X (d, s) have a nonempty intersection. Namely, if d is odd, then X (d, 2) ∼ = DW(d, 3); see Lemma 2.4. Let K6 − 3K2 denote the complete graph on 6 vertices with a 1-factor removed, and observe that K6 − 3K2 ∼ = W(3, 2), which, in turn, is isomorphic to the octahedron graph, and also to the line graph of the complete graph K4 . Further, let C3 C3 be the Cartesian product of two copies of the cycle C3 on 3 vertices (see [7] for the definition of the Cartesian product of graphs). We can now state the main result of the paper.

272

P. Potočnik / European Journal of Combinatorics 36 (2014) 270–281

Theorem 1.5. Let Γ be a tetravalent graph of order n admitting an arc-transitive group of automorphisms G such that the vertex-stabiliser Gv is isomorphic to the Klein group Z22 . Suppose that there exists a Gconsistent cycle in Γ of length r with r > n/2. Then one of the following holds: (i) Γ ∼ = K6 − 3K2 , r = 4 and |Aut(Γ ) : G| = 2; (ii) Γ ∼ = Cir(n, {1, a, −a, −1}) for some element a ∈ Z∗n \ {1, −1}, satisfying a2 = 1, r = n and G = Aut(Γ );   (iii) n is divisible by 4, Γ ∼ = W 2n , 2 , r = n and G is a group described in Lemma 2.1; (iv) n is an odd multiple of 3, Γ ∼ = DW |Aut(Γ ) : G| = 2).

n

3

 ,3 ,r =

2n 3

and G = Aut(Γ ) (unless n = 9, in which case

We shall prove Theorem 1.5 in Section 3, while Section 2 is devoted to the detailed analysis of the graphs W(m, 2), X (d, s) and DW(d, 3). 2. Exceptional families 2.1. The wreath graphs Let Γ = W(m, 2) for some m, m ≥ 3. For i ∈ Zn , let Bi = {(i, 0), (i, 1)}, let B = {Bi : i ∈ Zn }, and observe that B is a partition of V(Γ ) into sets of size 2 (see Definition 1.3). Note that every permutation of V(Γ ) that preserves the partition B is in fact an automorphism of Γ . Let K denote the set of permutations of Γ the act upon B trivially. Then K is an elementary abelian 2-group generated by transpositions τi for i ∈ Zn , where τi swaps the two vertices in Bi and fixes every Bj , j ̸= i, point-wise. For X ⊆ Zn let

τX =



τi .

i∈X

Furthermore, let ρ and σ be the automorphisms of Γ permuting the vertices of Γ according to the rule

(i, j)ρ = (i + 1, j),

(i, j)σ = (−i, j),

for every (i, j) ∈ Zm × Z2 = V(Γ ), and let A = ⟨K , ρ, σ ⟩. Observe that A is permutation isomorphic to the imprimitive wreath product Dm wrSym(2). It is well known (and easy to see) that unless m = 4 (in which case Γ ∼ = K4,4 ), the group A coincides with Aut(Γ ) (see for example [17, Corollary 2.9]). Let us now state the main result of this section. Lemma 2.1. Assume the notation of the previous paragraph and let G be an arc-transitive group of automorphisms of Γ such that Gv ∼ = Z22 . Suppose that there exists a G-consistent cycle of Γ of length r with r ≥ m + 1. Then one of the following holds: (i) m = 3, r = 4 = 32 |V(Γ )| and G = ⟨τ0 τ1 , ρ, σ ⟩ with |Aut(Γ ) : G| = 2; (ii) m is even, r = 2m = |V(Γ )| and G is conjugate in Aut(Γ ) to the group

⟨τ2Zm , ρτ0 , σ τ0 ⟩. Proof. Denote the vertex (0, 0) of Γ by v . Recall that, unless m = 4, the partition B is invariant under Aut(Γ ). On the other hand, if m = 4, then Γ ∼ = K4,4 , and a computer-aided computation easily shows that Aut(Γ ) contains two conjugacy classes of arc-transitive subgroup G with the property that Gv ∼ = Z22 , and that both these two subgroups preserve the partition B . This implies that (even in the case when m = 4) G is a subgroup of A = ⟨K , ρ, σ ⟩, the largest subgroup of Aut(Γ ) that preserves B . Let C = v0 v1 v2 . . . vr −1 be a G-consistent cycle of Γ of length r with r ≥ m + 1, and let g ∈ G be a shunt of C . Suppose first that for some i ∈ Zr , the vertices vi−1 and vi+1 lie in the same block Bj for some j ∈ Zm . Then vi ∈ Bj±1 for some choice of the sign in the index j ± 1. Observe that the shunt g then swaps the

P. Potočnik / European Journal of Combinatorics 36 (2014) 270–281

273

blocks Bj and Bj±1 , implying that the orbit of vi under the group ⟨g ⟩ is contained in Bj ∪ Bj±1 . On the ⟨g ⟩

other hand, the orbit vi is the set {v0 , v1 , . . . , vr −1 }, showing that r ≤ 4. However, since r ≥ m + 1 and m ≥ 3, this implies that m = 3 and r = 4. Note that then Γ = W(3, 2), which is isomorphic to the complete graph K6 with a 1-factor 3K2 removed. Hence |Aut(Γ )| ∼ = |Aut(3K2 )| = 6 · 23 . On the other hand, |G| = 6|Gv | = 6 · 22 , implying that |Aut(Γ ) : G| = 2. It is now easy to see (either by hand or by Magma [3]) that the only arc-transitive subgroup of Aut(Γ ) of index 2 with the property that Gv ∼ = Z22 is the group G = ⟨τ0 τ1 , ρ, σ ⟩. We may thus assume that for every i ∈ Zr , the vertices vi−1 and vi+1 lie in distinct blocks from B . Note that, the cycles of Γ with this property are either of length m (in which case the cycle contains exactly one vertex from every block in B ) or of length 2m. Since r ≥ m + 1, we thus see that r = 2m. In particular, a shunt of C has order 2m and acts upon B as a cyclic rotation Bi → Bi+1 . It is easy to see that every such automorphism of Γ is of the form ρτX for some X ⊆ Zm . For X ⊆ Zn and i ∈ Zn , let X + i = {x + i : x ∈ X } and let −X = {−x : x ∈ X }. Note that |X |

(ρτX )m = ρ m τX τX +1 . . . τX +m−1 = τZm . In particular, |X | is odd. Now observe that the mapping K → K , τY → τY τY +1 , is a group homomorphism whose kernel is the group of order 2 generated by τZn . The image of this homomorphism is thus of index 2 in K . Since every element τZ of that image has the property that |Z | is even, this implies that the set of elements of the form τY τY +1 is precisely the set of all elements τZ with |Z | even. Hence, since |X | is odd, there exists some Y ⊆ Zn such that τX = τ0 τY τY +1 . Finally, observe that

(ρτX )τY = τY ρτY τX = ρτY τY +1 τX = ρτ0 . We have thus shown that ρτX is conjugate by some element of K to the automorphism ρτ0 . We shall thus assume henceforth that ρτ0 ∈ G. Now consider the quotient graph Γ /B , that is, the graph with vertex-set B and an edge between two blocks B, C ∈ B whenever there is an edge between a vertex in B and a vertex in C . Note that Γ /B is a cycle of length m. Since G acts transitively on the arcs of Γ and G ∩ K is the kernel of the action of G on B , the quotient group G/(G ∩ K ) acts faithfully (in an obvious way) on Γ /B as an arc-transitive group of automorphisms. However, Γ /B is a cycle of length m, and thus G/(K ∩ G) ∼ = Dm . Since every preimage of the reflection about the vertex B0 of Γ /B is of the form σ τD , we may thus conclude that G = ⟨G ∩ K , ρτ0 , σ τD ⟩, for some D ⊆ Zm . Let us now try to determine the group G ∩ K . Note first that |G| = |G ∩ K ||Dn | = 2n|G ∩ K |. Hence, since Gv ∼ = Z22 , we see that |G| = 4 · |V(Γ )| = 8n, and thus |G ∩ K | = 4. Further, note that (ρτ0 )m acts upon B trivially, implying that (ρτ0 )m ∈ G ∩ K . However, (ρτ0 )m = τZm , showing that τZm ∈ G ∩ K . Let τ be a non-trivial element of G ∩ K other than τZm . Since ρ normalises G ∩ K and centralises τZm , we see that the conjugation by ρ swaps the elements τ and τ τZm (in particular, the order m of ρ is even), implying that τ τ ρ = τZm . It is now easy to see that τ = τ2Zm , implying that G ∩ K = {1, τ2Zm , τ2Zm +1 , τZm }, and hence G = ⟨τ2Zm , ρτ0 , σ τD ⟩. To complete the proof, we now show that D can be chosen to be {0}. First observe that since G ∩ K is non-trivial, we may choose D in such a way that 0 ∈ D. Then the element σ τZm τD fixes the vertex v and belongs to G. Since Gv ∼ = Z22 , this implies that σ τZm τD has order 2, and therefore (σ τZm τD )2 = τD τ−D = 1. In particular, we see that D = −D. Further, note that (ρτ0 σ τD )2 fixes every element of B set-wise, implying that (ρτ0 σ τD )2 ∈ G ∩ K . On the other hand, (ρτ0 σ τD )2 = τ0 τ−1 τ−D−1 τD , and since D = −D, we see that τ0 τ−1 τ−D−1 τ−D ∈

274

P. Potočnik / European Journal of Combinatorics 36 (2014) 270–281

K ∩ G. Finally, since K ∩ G is normalised by σ , and since τ0 τ1 τD τD+1 = (τ0 τ−1 τ−D−1 τ−D )σ , we see that τ0 τD τ1 τD+1 ∈ G ∩ K . By letting C = D ∪ {0} and recalling that D = −D, we thus see that C satisfies the following conditions: 0 ̸∈ C ,

C = −C ,

τC τC +1 ∈ G ∩ K ;

(∗)

in particular, τC τC +1 ∈ {1, τ2Zm , τ2Zm +1 , τZm }. Recall that the mapping K → K , τY → τY τY +1 , is a group homomorphism whose kernel is the group ⟨τZm ⟩; in particular, if we find one solution C of the equation τC τC +1 = τ for a fixed τ ∈ G ∩ K , then Zm \ C is the other (and exactly one of these two solutions will satisfy the condition 0 ̸∈ C ). If τC τC +1 = 1, then C = ∅ or C = Zm and since 0 ̸∈ C , we see that C = ∅ and D = {0}; implying that G is as in part (ii). If τC τC +1 = τZm , then only C = 2Zm + 1 satisfies the conditions (∗). Hence G = ⟨τ2Zm , ρτ0 , σ τ0 τ2Zm +1 ⟩. Since τ2Zm +1 ∈ G ∩ K , it follows that G = ⟨τ2Zm , ρτ0 , σ τ0 ⟩, as in part (ii) of the lemma. If τC τC +1 = τ2Zm , then m is divisible by 4 (since |2Zm | has to be even for a solution to exist) and the two solutions are C = 4Zm ∪ (4Zm + 1) and C = 4Zm+2 ∪ (4Zm+3 ). None of them satisfies the condition C = −C , a contradiction. Finally, if τC τC +1 = τ2Zm +1 , then again m is divisible by 4 and the C is either C = 4Zm ∪ (4Zm + 3) or C = (4Zm + 1) ∪ (4Zm + 2), again, none of these two solutions satisfies C = −C . Hence this case does not occur.  2.2. Depleted wreath graphs Recall that DW(d, 3) is isomorphic to the direct product Cd × K3 and as such admits a group of automorphisms isomorphic to Aut(Cd ) × Aut(K3 ) ∼ = Dd × S3 . This group is in fact generated by the automorphisms ρ, α, σ and τ of DW(d, 3) defined by:

(x, y)ρ = (x + 1, y), (x, y)α = (−x, y),  (x, 2); if y = 2 (x, y)τ = (x, 0); if y = 1 (x, 1); if y = 0.

(x, y)σ = (x, y + 1),

Let G = ⟨ρ, α, σ , τ ⟩ and observe that G is indeed isomorphic to Dd × S3 . Moreover, G acts transitively on the arcs of DW(d, 3) and Gv ∼ = Z22 for every vertex v of DW(d, 3). As we shall now see, with the exception of some small cases, the group G is in fact the full automorphism group of DW(d, 3). Lemma 2.2. Let Γ = DW(d, 3) and let G be as above. Then Gv ∼ = Z22 and the following hold: (i) (ii) (iii) (iv)

if if if if

d d d d

= 3, then Γ ∼ = C3 C3 and |Aut(Γ ) : G| = 2; = 4, then Γ ∼ = W(6, 2) and |Aut(Γ ) : G| = 16; = 6, then Γ ∼ = DW(3, 3) × K2 ∼ = (C3 C3 ) × K2 and |Aut(Γ ) : G| = 2; = 5 or d ≥ 7, then Aut(Γ ) = G.

Proof. If d ≤ 6, then the claim of the lemma can be easily checked by hand or by Magma [3]. Let us thus assume that d ≥ 7. Since G already acts regularly on the arcs of Γ , it suffices to show that any automorphism of Γ that fixes an arc is trivial. A typical edge of Γ is of the form {(i − 1, j − s), (i, j)} for some i ∈ Zd , j ∈ Z3 , s ∈ Z3 \ {0}. Let us call that edge e. Observe that out of the three neighbours of (i, j) that are not equal to (i − 1, j − s) only (i + 1, j + s) does not lie on a 4-cycle through e. This implies that every automorphism that fixes an underlying arc of e fixes also (i + 1, j + s), and by applying this argument repeatedly, it fixes every vertex of the form (i + k, j + ks) for k ∈ Z. If d is not divisible by 3, every vertex of Γ is of that form, showing that the stabiliser of an arc underlying e is trivial, which concludes the proof of the lemma. We may thus assume that d is divisible by 3.   For k ∈ Zd let xk denote the arc (i + k, j + ks), (i + k + 1, j +(k + 1)j) and let C be the directed cycle of Γ consisting of the arcs xk , k ∈ Zd . Since d is divisible by 3, we see that the length of C is d. Note that the automorphism h = ρσ s rotates C one step forward. The previous paragraph shows that an automorphism of Γ that fixes one arc of C fixes every arc of C . More generally, if an automorphism g

P. Potočnik / European Journal of Combinatorics 36 (2014) 270–281

275

of Γ maps an arc x0 to an arc xk for some k, then hk g −1 fixes x0 and thus fixes C point-wise. This implies that g fixes C set-wise. We have thus shown that C is a block of imprimitivity for the action of Aut(Γ ) on the arcs of Γ . But then the underlying (undirected) cycle of C is clearly a block of imprimitivity for the action of Aut(Γ ) of the edges of Γ . In other words, The set C of G-images of that undirected cycle forms an Aut(Γ )-invariant partition of E (Γ ). A simple counting argument shows that |C | = 6. Observe that there are some obvious 6-cycles in Γ , namely those induced by the set of vertices of the form (i, j) and (i+1, j′ ) for a fixed i and arbitrary j, j′ ∈ Z3 . Let us call these 6-cycles trivial. Note that each trivial 6-cycle intersects every member of C exactly once. On the other hand, a straightforward computation shows that no non-trivial 6-cycle has this property; we leave the details to the reader. Since C is an Aut(Γ )-invariant partition of E (Γ ), a trivial 6-cycle therefore cannot be mapped to a non-trivial 6-cycle by an automorphism of Γ . In particular, an automorphism of Γ that fixes an arc of a trivial 6-cycle fixes each arc of that 6-cycle. Since a trivial 6-cycle intersects each cycle in C , this implies that an automorphism fixing an arc of a trivial 6-cycle fixes one arc in every element of C . But since an automorphism fixing an arc of a member of D ∈ C fixes every arc of D and since every arc of Γ is contained in some element of C , this shows that an automorphism that fixes an arc of a trivial 6-cycle is trivial. In particular, Aut(Γ ) acts regularly on the arcs of Γ and hence Aut(Γ ) = G.  Remark. Note that under additional assumption of d being odd, we could prove the above result by using the well-known result on the automorphism group of the direct product of non-bipartite graphs [7, Theorem 8.18]. 2.3. The graphs X (d, s) This section is devoted to the analysis of the graph X (d, s). In a series of lemmas, we will show that a graph X (d, s) admits a group of automorphisms having two orbits on the set of vertices, two orbits on the set of edges and two orbits on the set of arcs. We will then characterise those graphs X (d, s) that are arc-transitive and study their properties in some detail. In the end, our analysis will amount to the following theorem. Theorem 2.3. Let d and s be integers satisfying 1 ≤ s < d. Then the graphs X (d, s) and X (d, d − s) are isomorphic. Moreover, the graph X (d, s) is arc-transitive if and only if (d, s) = (2, 1) (in which case the graph X (d, s) is isomorphic to W(3, 2)), or d is odd and s = 2 or s = d − 2 (in which case X (d, s) is isomorphic to the depleted wreath graph DW(d, 3)). Moreover, if X (d, s) is arc-transitive, then the automorphism group of X (d, s) contains an arc-transitive subgroup G with Gv ∼ = Z22 . Let us start with a lemma which establishes a link between the graphs X (d, 2) and the depleted wreath graphs. Lemma 2.4. Let d be an odd integer, d ≥ 3. Then X (d, 2) ∼ = DW(d, 3). Proof. Let ϕ : V (X (d, 2)) → Zd × Z3 be the mapping defined by ui → (i + 1 mod d, 0),

vi → (i mod d, (−1)i mod 3)

for i ∈ {0, 1, . . . , 2d − 1}. It is a matter of straightforward computation to see that this is an isomorphism between the graphs X (d, 2) and DW(d, 3); see also Fig. 1.  A graph X (d, s) is clearly connected, has order 3d and is regular of valency 4. Throughout the rest of the section, let d and s be arbitrary integers satisfying the conditions in Definition 1.4. In fact, in view of the following lemma, we shall also assume that 1 ≤ s ≤ d/2. Lemma 2.5. The graphs X (d, s) and X (d, d − s) are isomorphic. Proof. Observe that the mapping which maps vj in X (d, s) to vd−j in X (d, d − s) and ui in X (d, s) to ud−i in X (d, d − s) (indices at vj and uj computed modulo 2d and d, respectively) is an adjacency preserving bijection, and thus an isomorphism between the two graphs. 

276

P. Potočnik / European Journal of Combinatorics 36 (2014) 270–281

Fig. 1. The graph X (d, 2), d odd, as DW(d, 3).

Let us now examine some obvious symmetries that a graph X (d, s) (call it Γ ) possesses. Let ρ, σ and τ be the permutations of V (Γ ) defined by the following rules:

ρ : vi →  vi+1 ρ : ui →  ui+1

σ : vi → vd+i τ : vi → vs−i σ : ui → ui τ : ui → u−i

where all the indices at vi are computed modulo 2d and those at ui modulo d. Note that ρ and τ preserve adjacency in Γ , and that σ = ρ d , implying that all three permutations are automorphisms of Γ . Furthermore, note that (ρτ )2 = 1, showing that the group H = ⟨ρ, τ ⟩

(1)

is isomorphic to the dihedral group D2d of order 4d, with ⟨σ ⟩ being the centre of H. Let W = {vi : i ∈ {0, . . . , 2d − 1}}, let U = {ui : i ∈ {0, . . . , d − 1}} and let EW (EU , respectively) denote the set of edges having both endpoints in W (one endpoint in U and one in W , respectively). It is now easy to see that the following holds: Lemma 2.6. Let Γ = X (d, s) and let H be as in (1). Then H has two orbits on V (Γ ) (namely W and U) and two orbits on E (Γ ) (namely EU and EW ). The vertex-stabiliser Hu0 is isomorphic to the Klein group Z22 and acts transitively on the neighbourhood of u0 . The graph Γ is arc-transitive if and only if there exists an automorphism of Γ mapping a vertex from U to a vertex in W . Proof. To prove that W and U are orbits of H, observe that they are already orbits of ⟨ρ⟩ and that σ preserves both W and U set-wise. Similarly, H clearly preserves the sets EW and EU and, due to the generator ρ , acts transitively on EW . Since |H | = 4|U |, the stabiliser Hu0 has order 4. On the other hand, σ and τ both fix u0 , implying that Hu0 = ⟨σ , τ ⟩. The latter is clearly isomorphic to the Klein group Z22 and acts transitively on the neighbourhood of u0 . This implies that H acts transitively on EU , showing that EW and EU are indeed the two orbits of H in E (Γ ). If Γ is arc-transitive, then it is also vertex-transitive, implying that there is an automorphism mapping a vertex from U to a vertex in W . Conversely, if there existed an automorphism α of Γ mapping a vertex from U to a vertex of W , then the group G = ⟨H , α⟩ would act transitively on V (Γ ). Moreover, the stabiliser Gw of a vertex w ∈ W would then act transitively on the neighbourhood Γ (w), making the graph Γ locally G-arctransitive. Since G is also vertex-transitive, this would then imply that Γ is G-arc-transitive.  Let us now have a look at the smallest representatives of the family, namely, the graphs X (2, 1), X (3, 1), X (4, 1) and X (4, 2). Since these graphs are rather small, it is not difficult to see (either by hand or by Magma) that the following lemma holds. Lemma 2.7. The graphs X (2, 1) and X (3, 1) are isomorphic to the wreath graph W(3, 2) and the Cartesian product C3 C3 (which is isomorphic to DW(3, 3)), respectively. The graphs X (4, 1) and X (4, 2) are not arc-transitive. In view this lemma, we shall assume henceforth that Γ = X (d, s) for some integers d and s satisfying d ≥ 5 and 1 ≤ s ≤ d/2.

P. Potočnik / European Journal of Combinatorics 36 (2014) 270–281

277

Fig. 2. 4-cycles in X (d, s) passing through u0 vs .

Fig. 3. 4-cycles in X (d, s) passing through vs vs+1 .

The next few paragraphs are devoted to the proof that Γ can only be arc-transitive if s = 2. To this end, we shall determine the number of 4-cycles that pass through a given edge e of Γ ; we shall denote this number c4 (e). Of course, this number is constant within an edge-orbit of Aut(Γ ) hence it suffices to consider an edge from EU , say u0 vs , and an edge from EW , say vs vs+1 . Note that

Γ (ui ) = {vi , vi+s , vd+i , vd+i+s } and Γ (vi ) = {vi−1 , vi+1 , ui−s , ui }.

(2)

To determine the number c4 (u0 vs ) of 4-cycles passing through u0 vs , we need to count the number of edges having one endvertex in Γ (u0 ) \ {vs } and the other in Γ (vs ) \ {u0 }. Observe that v0 ∈ Γ (u0 ) \ {vs } has no neighbours in Γ (vs ) \ {u0 }, unless s = 2, in which case v0 is adjacent to vs−1 = v1 , or 2s = d, in which case v0 is adjacent to u−s = us ; see Fig. 2. Similarly, vd ∈ Γ (u0 ) \ {vs } has no neighbours in Γ (vs ) \ {u0 }, unless 2s = d, in which case it is adjacent to ud−s = us . Finally, the only element of Γ (vs ) \ {u0 } that is adjacent to vd+s is us . To summarise, we have the following possibilities (note that s = 2 and d = 2s cannot occur simultaneously since we are assuming that d ≥ 5): (i) if s ̸= 2 and d ̸= 2s, then c4 (u0 vs ) = 1; (ii) if d = 2s, then c4 (u0 vs ) = 3 and the three 4-cycles containing u0 vs intersect in the 2-path u0 vs us ; (iii) if s = 2, then c4 (u0 vs ) = 2 and the two 4-cycles containing u0 vs intersect only in u0 vs . Let us now determine the number c4 (vs vs+1 ) of 4-cycles passing through vs vs+1 . To do that, we need to count the number of edges between the sets Γ (vs ) \ {vs+1 } and Γ (vs+1 ) \ {vs }. Similarly as above, a straightforward computation (see also Fig. 3) shows that such an edge exists if and only if s = 2, in which case there are two such edges: vs−1 u1 and us vs+2 . Hence we have the following: (i) if s ̸= 2, then c4 (vs vs+1 ) = 0; (ii) if s = 2, then c4 (u0 vs ) = 2 and the two 4-cycles containing vs vs+1 intersect only in vs vs+1 .

278

P. Potočnik / European Journal of Combinatorics 36 (2014) 270–281

Since the number c4 (e) is the same for every e ∈ E (Γ ) whenever Γ is arc-transitive, we have thus proved the following: Lemma 2.8. Let Γ = X (d, s) for some integers d and s satisfying d ≥ 5 and 1 ≤ s ≤ d/2. If Γ is arc-transitive, then s = 2. If s = 2, then every edge e of X (d, s) lies on precisely two 4-cycles and these two 4-cycles intersect only in e. Let us now restrict ourselves to the family of graphs X (d, 2) with d ≥ 5. In the following paragraphs, we shall use some basic concepts from the topological graph theory, in particular, the theory of graph embeddings and maps on surfaces. For the missing definitions, we refer the reader to [6]. Let us think of X (d, 2) as a 1-dimension complex in the usual manner, where 0-cells are the vertices and 1-cells are the edges of the graph. By gluing a 2-cell along the boundary of each 4-cycle, one obtains a 2-dimensional complex S whose 1-dimensional skeleton is X (d, 2). Since each edge lies on precisely two 4-cycles, each point in the interior of an edge will have an Euclidean neighbourhood in S (that is, a topological neighbourhood in S homeomorphic to an open disc). Similarly, since the two 4-cycles containing the same edge meet in that edge only, it is not difficult to see that every vertex of the graph will have an Euclidean neighbourhood in S . This implies that S is in fact a closed surface. The embedding of X (d, 2) into S gives rise to a map, call it M (d), with all faces of length 4 and the underlying graph being X (d, 2). Maps in which every face has length m and every vertex has valence n are said to have type {m, n}. The map M (d) is thus of type {4, 4}. The Euler formula now implies that the underlying surface M is either the torus (if M is orientable) or the Klein bottle (if M is nonorientable). Finally, since every automorphism of X (d, 2) preserves the set of 4-cycles, it also extends to an automorphism of M (d). We can summarise the above in the following lemma: Lemma 2.9. The map M (d), obtained by gluing a disc along the boundary of each 4-cycle in X (d, 2), is a map of type {4, 4} on the torus or the Klein bottle. Every automorphism of X (d, 2) extends to an automorphism of M (d). Let us now show that the underlying surface of M (d) is the Klein bottle whenever d is even. To see this, observe first that the graph X (d, 2) can be obtained from the graph X ′ (d, 2) on 4d vertices, denoted v0 , . . . , v2d−1 and u′0 , . . . , u′2d−1 , with edges of the form vi vi+1 , u′i vi and u′i vi+1 , for i ∈ {0, . . . , 2d − 1}, after identifying the vertex u′i with u′i+d (and thus forming the vertex ui of X (d, 2)) for each i ∈ {0, . . . , 2d − 1}; note that all the indices in this definition should be taken modulo 2d. The graph X ′ (d, 2) can be drawn in a ring R bounded by two concentric circles (C1 and C2 ), in such a way that the vertices vi are placed equidistantly onto a circle between C1 and C2 , the vertices u′i with even i are placed equidistantly on C1 and the vertices u′i with odd i are placed equidistantly on C2 . The graph X (d, 2) and the surface S underlying the corresponding map M (d) is then obtained from the ring R by identifying pairs of antipodal points on C1 and on C2 , respectively (see Fig. 4). The surface obtained from R in such a way is clearly not orientable and is in fact isomorphic to the Klein bottle. Maps on the Klein bottle admitting certain symmetry have been studied in several papers. For example, it was shown in [20] (see also [15]) that the underlying graph of an edge-transitive map on the Klein bottle of type {4, 4} is necessarily isomorphic to the wreath graph W(n, 2). Since the graphs X (d, 2), for d ≥ 3, are clearly not isomorphic to the wreath graphs, Lemma 2.9 implies the following: Lemma 2.10. If d is an even integer greater than 5, then M (d) is a map on the Klein bottle and X (d, 2) is not arc-transitive. Proof. If d ≥ 6 is even, then, in view of the above, the map M (d) in not edge-transitive. However, in view of Lemma 2.9, M (d) is edge-transitive whenever X (d, 2) is edge-transitive. Hence X (d, 2) is not edge-transitive (and thus neither arc-transitive).  We are now ready to prove Theorem 2.3. Let d and s be integers satisfying 1 ≤ s < d. Then the graphs X (d, s) and X (d, d − s) are isomorphic by Lemma 2.5. Let Γ = X (d, s). Suppose first that (d, s) = (2, 1) or d is odd and s ∈ {2, d − 2}. We need to show that, in this case, Γ admits an arc-transitive group G with Gv ∼ = Z22 .

P. Potočnik / European Journal of Combinatorics 36 (2014) 270–281

279

Fig. 4. The graph X (d, 2), d even, embedded on the Klein bottle.

If (d, s) = (2, 1), then Γ is isomorphic to W(3, 2) by Lemma 2.7. Note that in this case Aut(Γ ) is isomorphic to S4 × C2 and contains a subgroup G of index 2 (not isomorphic neither to S4 nor to A4 × C2 ), which acts arc-transitively on Γ and such that Gv ∼ = Z22 . If d is odd and s ∈ {2, d − 2}, then, by Lemmas 2.5 and 2.4, Γ is isomorphic to DW(d, 3). Furthermore, in view of Lemma 2.2, Γ admits an arc-transitive group G with Gv ∼ = Z22 . Conversely, suppose now that Γ is arc-transitive. To conclude the proof of Theorem 2.3, we need to show that (d, s) = (2, 1) or d is odd and s ∈ {2, d − 2}. In view of Lemma 2.5, we may (and shall) assume that s ≤ d/2 and show that (d, s) = (2, 1) or (d, s) = (3, 1) or d is odd and s = 2. If d = 2 or d = 3, then condition s ≤ d/2 automatically implies that s = 1, and the claim holds. If d = 4, then s = 1 or s = 2, but then, by Lemma 2.7, Γ is not arc-transitive, which is a contradiction. We may thus assume that d ≥ 5. In view of Lemma 2.8, it follows that s = 2. If d is even, then by Lemma 2.10, Γ is not arc-transitive, a contradiction. Hence d is odd and Theorem 2.3 is proved. 3. Proof of Theorem 1.5 This section is devoted to the proof of Theorem 1.5. Let Γ be a tetravalent graph of order n admitting an arc-transitive group of automorphisms G such that Gv ∼ = Z22 . Let C = v0 v1 v2 . . . vℓ−1 be a Gg consistent cycle of Γ and let g ∈ G be a shunt for C (i.e. vi = vi+1 for all i ∈ Zℓ ). Suppose that the length ℓ of C is strictly more than n/2. In order to prove Theorem 1.5, we need to show that one of the following occurs: (i) Γ ∼ = K6 − 3K2 , ℓ = 4 and |Aut(Γ ) : G| = 2; ∼ Cir(n, {1, a, −a, −1}) for some element a ∈ Z∗ \ {1, −1}, satisfying a2 = 1, ℓ = n and (ii) Γ = n G = Aut(Γ ); n  (iii) n is divisible by 4, Γ ∼ = W 2 , 2 , ℓ = n and G is a group described in Lemma 2.1; (iv) n is an odd multiple of 3, Γ ∼ = DW |Aut(Γ ) : G| = 2).

n

3

 ,3 ,ℓ =

2n 3

and G = Aut(Γ ) (unless n = 9, in which case

Suppose first that ℓ = n. Then Γ ∼ = Cir(n; S ) for some S containing 1. Arc-transitive circulants have been classified independently by Kovács [9] and Li [12]. It follows from their work (see, for example,

280

P. Potočnik / European Journal of Combinatorics 36 (2014) 270–281

[9, Theorem 1]) that either Γ ∼ = K5 , or Γ ∼ = W(m, 2) for m = 2n , or Γ is a normal circulant (see [21] for definition of a normal Cayley graph), or n = 3d for some integer d and Γ ∼ = DW(d, 3). Note that if Γ ∼ = K5 , then G is a doubly-transitive group of degree 5. It is easy to see that then G ∼ = Alt(5), Sym(5) or the Frobenius group Z5 o Z4 . None of these groups have vertex-stabiliser isomorphic to Z22 ; a contradiction. If Γ ∼ = W(m, 2), then by Lemma 2.1, (iii) holds. If Γ is a normal circulant, then by definition, Aut(Γ ) contains a normal cyclic subgroup isomorphic to Zn . In particular, we may assume that Γ = Cir(n, S ) for some S ⊆ Zn \ {0} satisfying S = −S. Let A = Aut(Γ ) and consider the vertex-stabiliser A0 of the vertex 0 of the circulant Cir(n, S ). It is well known (see for example [21, Proposition 1.5]) that since Zn is normal in A, the group A is a semidirect product of Zn (acting regularly on V (Cir(n, S )) = Zn by addition) by A0 (acting on Zn as a group of automorphisms of Zn ). In particular, A0 can be viewed as a subgroup of the multiplicative group Z∗n Γ (0)

acting on Zn by multiplication. Since Z∗n is abelian, so is A0 , implying that the permutation group A0 , induced by the action of A0 on the neighbourhood of 0, is regular. Using the connectivity of Γ , one then easily shows that A acts regularly on the arcs of Γ ; in particular, A = G and A0 ∼ = Z22 . Moreover, since the neighbourhood of 0 in Cir(n, S ) is S, it follows that S = bA0 for any b ∈ S. Since S has to generate Zn , it follows that b is a generator of Zn . By applying an appropriate automorphism of Zn , we may thus assume that b = 1 and thus S = A0 . Since A0 is a subgroup of Z∗n isomorphic to Z22 , this shows that A0 = {1, −1, a, −a} for some involution a ∈ Z∗n . In particular, (ii) holds. To finish the case where ℓ = n, we may thus assume that n = 3d for some integer d and Γ ∼ = DW(d, 3). If d = 3, then Γ ∼ = DW(3, 3) ∼ = C3 C3 , and it can be seen easily that every consistent cycles of Γ is of length 3, 4 or 6, contradicting the assumption that ℓ = n. Further, if d = 4, then by Lemma 2.2, Γ ∼ = W(6, 2) and by Lemma 2.1, (iii) holds. Similarly, if d = 6, then Γ ∼ = (C3 C3 ) × K2 , which can be shown to have only consistent cycles of lengths 4 and 6; a contradiction. We may thus assume that d = 5 or d ≥ 7, which in view of Lemma 2.2 implies that Aut(Γ ) = G ∼ = Dd × S3 . Since g is a shunt of a cycle of length n, the order of g is n, and thus ⟨g ⟩ is a cyclic subgroup of Dd × S3 of order n = 3d. However, the group Dd × S3 contains such a cyclic group only when d is not divisible by 3; in the latter case this subgroup is unique and thus normal. Hence, the graph Γ is a normal circulant and by the above, (ii) holds. This finishes the case where ℓ = n. Suppose through the rest of the section that n/2 < ℓ < n. Consider the subgraph ∆ of Γ induced by the set V (C ). This graph is clearly a circulant with respect to the cyclic group ⟨g ⟩ and has valence at least 2. Furthermore, since ℓ < n, there exists at least one vertex in V (Γ ) \ V (C ) which is adjacent to a vertex in C . Hence the valence of ∆ is at most 3. If the valence of ∆ is 3, then ∆ is a cubic circulant, implying that ℓ is even and that v0 is adjacent to vℓ/2 . Observe also that the number of edges with one endvertex on C and one outside C is precisely ℓ (there is one such edge at every vertex in C ). Since |V (Γ ) \ V (C )| < ℓ, it follows that there is a vertex v ∈ V (Γ ) \ V (C ) which is adjacent to at least two vertices of C . Without loss of generality we may assume that one of these two vertices is v0 and let vr be another neighbour of v in C . Note that v is the only neighbour of v0 and of vr that lies outside C . On the other r r hand, if we apply g r to Γ , we see that vr is adjacent also to v g , implying that v = v g , and thus g r ∈ Gv . r Since the exponent of the stabiliser Gv is 2, it follows that g is an involution, and therefore ℓ = 2r. In particular, the vertex v has precisely two neighbours in C , namely v0 and vℓ/2 . This gives rise to a cycle v0 vvr of length 3 in Γ . Tetravalent arc-transitive graphs of girth 3 have been classified in [16], where it was proven that such a graph is isomorphic to the line graph of a cubic 2-arc-transitive graph. In particular, n = 3t /2 for some even integer t (the order of the cubic 2-arc-transitive graph), and thus n is divisible by 3. Since Γ is assumed to be arc-transitive, the edge v0 v1 also lies on a cycle of length 3. Clearly v0 and v1 have no common neighbour outside C and can have a common neighbour on C only if ℓ = 4 (in which case that common neighbour is v2 = vℓ/2 ). Hence ℓ = 4 and thus n < 8. Since n is divisible by 3, we therefore have n = 6 and Γ must then be isomorphic to the line graph of a cubic 2-arctransitive graph on four vertices. The only such graph is the complete graph K4 , and its line graph is the octahedron, which is isomorphic to the wreath graph W(3, 2), which, in turn, is isomorphic to the graph K6 − 3K2 ; in particular, (i) holds in this case. This completes the case where the valence of ∆ is 3.

P. Potočnik / European Journal of Combinatorics 36 (2014) 270–281

281

We may thus assume that the valence of ∆ is 2. Then each vertex of C is adjacent to two distinct vertices in V (Γ ) \ V (C ). Let v be an arbitrary neighbour of v0 that does not lie on C and let Ω = v ⟨g ⟩ . Since |V (Γ ) − V (C )| < ℓ = |⟨g ⟩| it follows that the action of ⟨g ⟩ on Ω is not faithful. Let ⟨g d ⟩, where d is a divisor of ℓ, be the generator of the kernel of this action. Since g d ∈ Gv and since the exponent of Gv is 2, it follows that ℓ is even, d = ℓ/2 and that v is adjacent to v0 as well as to vd . Observe also that |Ω | = |⟨g ⟩|/|⟨g d ⟩| = ℓ/2. Let u and w be the two neighbours of v0 not lying on C . As we have seen above, both u and w are adjacent to v0 and vd and are fixed by the element g d . Then |V (C )|+|w ⟨g ⟩ |+|u⟨g ⟩ | ≥ ℓ+ℓ/2 +ℓ/2 > n, s implying that w ⟨g ⟩ = u⟨g ⟩ . Let s be the smallest positive integer such that u = w g . Then u is adjai

cent also to vs and vd+s . Now let ui = ug for i ∈ {0, . . . , d − 1} and observe that for arbitrary i ∈ {0, . . . , d − 1}, the vertex ui is adjacent to vi , vi+s , vi+d and vi+d+s . Every ui is thus adjacent to four vertices on C and every vertex vi is adjacent to two vertices on C and two vertices in U = {ui : i ∈ {0, . . . , d − 1}}, namely the vertices ui and ui−s with indices computed modulo d, this implies that V (Γ ) = V (C ) ∪ U, and in particular, the graph Γ is isomorphic to the graph X (d, s) from Definition 1.4. Note also that n = 3d, and since d = 2ℓ , we indeed see that the length ℓ of the consistent

, as stated in (iv). By Theorem 2.3, the graph X (d, s) is then either isomorphic to the graph cycle C is 2n 3 W(3, 2) ∼ = K6 − 3K2 (in which case (i) holds) or to the graph DW(d, 3) for some odd d (and thus (iv) holds). In the latter case, the claim about the automorphism group of Γ follows directly from Lemma 2.2. In particular, (iv) holds in this case. This completes the proof. Acknowledgements I am grateful to the referees whose valuable comments helped me to improve the manuscript considerably. References [1] N. Biggs, Aspects of symmetry in graphs, in: Algebraic Methods in Graph Theory, Vol. I, II (Szeged, 1978), in: Colloq. Math. Soc. János Bolyai, vol. 25, North-Holland, Amsterdam–New York, 1981, pp. 27–35. [2] M. Boben, Š. Miklavič, P. Potočnik, Consistent cycles in 12 -arc-transitive graphs, Electronic J. Combin. 16 (2009) #R5. [3] W. Bosma, J. Cannon, Handbook of Magma Functions, University of Sidney, 1994. [4] J.H. Conway, Talk Given at the Second British Combinatorial Conference at Royal Holloway College, Egham, 1971. [5] H.H. Glover, K. Kutnar, D. Marušič, Hamiltonian cycles in cubic Cayley graphs: the (2, 4k, 3) case, J. Algebraic Combin. 30 (2009) 447–475. [6] J. Gross, T.W. Tucker, Topological Graph Theory, Wiley, New York, 1987. [7] R. Hammack, S. Klavžar, W. Imrich, Handbook of Product Graphs, CRC Press, Boca Raton, 2011. [8] W.M. Kantor, Cycles in graphs and groups, Amer. Math. Monthly 115 (2008) 559–562. [9] I. Kovács, Classifying arc-transitive circulants, J. Algebraic Combin. 20 (3) (2004) 353–358. [10] K. Kutnar, I. Kovács, R. Janos, Rose window graphs underlying rotary maps, Discrete Math. 310 (2010) 1802–1811. [11] K. Kutnar, D. Marušič, A complete classification of cubic symmetric graphs of girth 6, J. Combin. Theory Ser. B 99 (2009) 162–184. [12] C.H. Li, Permutation groups with a cyclic regular subgroup and arc-transitive circulants, J. Algebraic Combin. 21 (2005) 131–136. [13] Š. Miklavič, P. Potočnik, S. Wilson, Consistent cycles in graphs and digraphs, Graphs Combin. 23 (2007) 205–216. [14] Š. Miklavič, P. Potočnik, S. Wilson, Overlap in consistent cycles, J. Graph Theory 55 (2007) 55–71. [15] A. Orbanić, D. Pellicer, T. Pisanski, T.W. Tucker, Edge-transitive maps of low genus, Ars Math. Contemp. 4 (2011) 385–402. [16] P. Potočnik, S. Wilson, Tetravalent edge-transitive graphs of girth at most 4, J. Combin. Theory Ser. B. 97 (2007) 217–236. [17] C.E. Praeger, M.Y. Xu, A characterization of a class of symmetric graphs of twice prime valency, European J. Combin. 10 (1989) 91–102. [18] J. Širáň, Regular maps on a given surface: a survey, Top. Discrete Math. Algorithms Combin. 26 (2006) 591–609. Part VI. [19] S. Wilson, Semi-transitive graphs, J. Graph Theory 45 (2004) 1–27. [20] S. Wilson, Uniform maps on the Klein bottle, J. Geom. Graph. 10 (2006) 161–171. [21] M.-Y. Xu, Automorphism groups and isomorphisms of Cayley digraphs, Discrete Math. 182 (1998) 309–319.