Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:3 https://doi.org/10.1007/s12044-018-0385-4 No hexavalent half-arc-transitive graphs of order twice a prime square exist MI-MI ZHANG Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China E-mail: [email protected] MS received 30 September 2016; revised 1 October 2017; accepted 3 November 2017; published online 19 March 2018 Abstract. Agraphishalf-arc-transitive if its automorphism group acts transitively on its vertex set and edge set, but not arc set. Let p be a prime. Wang and Feng (Discrete Math. 310 (2010) 1721–1724) proved that there exists no tetravalent half-arc-transitive graphs of order 2p2. In this paper, we extend this result to prove that no hexavalent half-arc-transitive graphs of order 2p2 exist. Keywords. Half-arc-transitive; bi-Cayley graph; vertex transitive; edge transitive. 2010 Mathematics Subject Classification. 05C25, 20B25. 1. Introduction All groups considered in this paper are finite, and all graphs are finite, connected, simple and undirected. For group-theoretic and graph-theoretic terminologies not defined here, we refer the reader to [4,24]. For a graph ,letV (), E(), A() and Aut () be the vertex set, the edge set, the arc set and the full automorphism group of , respectively. A graph is said to be vertex- transitive, edge-transitive or arc-transitive if Aut () acts transitively on V (), E() or A(), respectively. A graph is said to be half-arc-transitive provided that it is vertex- and edge-transitive, but not arc-transitive. The investigation of half-arc-transitive graphs was initiated by Tutte [20] who proved that a vertex- and edge-transitive graph with odd valency must be arc-transitive. In 1970, Bouwer [2] constructed an infinite family of half-arc-transitive graphs, and proved the existence of half-arc-transitive graphs of every even valency at least 4. After Bouwer’s work, many infinite families of half-arc-transitive graphs have been constructed (see, for example, [11,16–18,21]). In 1981, Holt [15] found a half-arc-transitive graph with 27 vertices and valency 4. Alspach et al. [1] proved that Holt’s graph is the smallest half-arc- transitive graph. In the literature, tetravalent half-arc-transitive graphs have also received a lot of attention (see [7,19]), and much work focuses on the classification problem. Let p beaprime.Xu[25] classified the tetravalent half-arc-transitive graphs of order p3 and Feng et al. [13] classified the tetravalent half-arc-transitive graphs of order p4.Itisknown that no half-arc-transitive graphs of order p or p2 exist (see [1]). In [8], Cheng and Oxley © Indian Academy of Sciences 1 3 Page 2 of 8 Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:3 proved that there exist no half-arc-transitive graphs of order 2p.In[23], Wang and Feng proved that no tetravalent half-arc-transitive graphs of order 2p2 exist. In this paper, we shall extend this result to the hexavalent half-arc-transitive graphs of order 2p2. The following is the main result of this paper. Theorem 1.1. No hexavalent half-arc-transitive graphs of order twice a prime square exist. 2. Preliminaries For a finite, simple and undirected graph and a subset B of V (), the subgraph of induced by B will be denoted by [B]. Let be a connected vertex-transitive graph, and let G ≤ Aut () be vertex-transitive on .ForaG-invariant partition B of V (),thequotient graph B is defined as the graph with vertex set B such that for any two vertices B, C ∈ B, B is adjacent to C if and only if there exist u ∈ B and v ∈ C which are adjacent in .LetN be a normal subgroup of G. Then the set B of orbits of N in V () is a G-invariant partition of V (). In this case, the symbol B will be replaced by N . Z Z∗ For a positive integer n,let n denote the cyclic group of order n and n be the mul- tiplicative group of Zn consisting of numbers coprime to n. For two groups M and N, N M denotes a semidirect product of N by M. For a subgroup H of a group G, denote by CG (H) the centralizer of H in G and by NG (H) the normalizer of H of G. Let G be a permutation group on a set and α ∈ . Denoted by Gα the stabilizer of α in G, that is, the subgroup of G fixing the point α. We say that G is semiregular on if Gα = 1 for every α ∈ and regular if G is transitive and semiregular. It is well-known that a graph is a Cayley graph if it has a group of automorphisms acting regularly on V (). If we, instead, require the graph to have a group of automorphisms acting semiregularly on V () with two orbits, then we obtain a so-called bi-Cayley graph. The following proposition is easy to obtain (see, for example, [23, Proposition 1.2]). PROPOSITION 2.1 Every edge-transitive Cayley graph over an abelian group is also arc-transitive. Note that every bi-Cayley graph admits the following concrete realization. Given a group H,letR, L and S be subsets of H such that R−1 = R, L−1 = L and R ∪ L does not contain the identity element of H.Thebi-Cayley graph over H relative to the triple (R, L, S), denoted by BiCay(H, R, L, S) is the graph having vertex set the union of the right part H0 ={h0 | h ∈ H} and the left part H1 ={h1 | h ∈ H}, and edge set the union −1 −1 of the right edges {{h0, g0}|gh ∈ R}, the left edges {{h1, g1}|gh ∈ L} and the −1 spokes {{h0, g1}|gh ∈ S}.If|R|=|L|=s, then BiCay(H, R, L, S)issaidtobe an s-type bi-Cayley graph (see [27]). Let = BiCay(H, R, L, S).Forg ∈ H, define a permutation R(g) on the vertices of by the rule R(g) = ( ) , ∀ ∈ Z , ∈ . hi hg i i 2 h H Then R(H) ={R(h) | h ∈ H} is a semiregular subgroup of Aut () which is isomorphic to H and has H0 and H1 as its orbits. When R(H) is normal in Aut (), the bi-Cayley graph Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:3 Page 3 of 8 3 = BiCay(H, R, L, S) will be called a normal bi-Cayley graph over H (see [28]). A bi-Cayley graph over an abelian group is simply called a bi-abelian. The following proposition is about half-arc-transitive bi-abelians. PROPOSITION 2.2 [9, Proposition 1.3] Every connected regular edge transitive bi-Cayley graph = BiCay(H, R, L, S) over an abelian group is vertex-transitive. Moreover, if is half-arc-transitive, then R ∪ Lis non-empty and does not contain involutions |R|=|L| is even, |S| > 2, and the valency of is at least 6. In view of [22], we have the following proposition. PROPOSITION 2.3 [22, Proposition 2.6] Let X be a connected half-arc-transitive graphs of valency 2n. Let A = Aut (X) and let Au be the stabilizer of u ∈ V (X) in A. Then each prime divisor of |Au| is a divisor of n!. In particular, if X has valency 6 then Au is a {2, 3}-group. By Theorems 1 and 3 of [5], we have the following proposition. PROPOSITION 2.4 [5] Let be a arc-transitive graph of order p and valency n with p a prime and 0 < n < p. Then n is even and divides p − 1 and |Aut ()|=np. Let p be a prime. The following two propositions are all cubic arc-transitive graphs of order 2p and 4p, respectively. PROPOSITION 2.5 [12, Proposition 2.8] Let p be a prime and let be a connected cubic arc-transitive graph of order 2p. Then is 1-,2-,3-or4-regular. Furthermore, −1 (1) is 1-regular if and only if is isomorphic to the graph Cay (D2p, {a, ab, ab }), p 2 where D2p =a, b | a = b = 1, bab = 1, p ≥ 13 and p − 1 is a multiple of 3.In −λ this case, the Cayley graph Cay (D2p, {a, ab, ab }) is normal and it is independent λ Z∗ of the choice of an element of order 3 in p. (2) is 2-regular if and only if is isomorphic to the complete graph K4 of order 4. (3) is 3-regular if and only if is isomorphic to the complete bipartite graph K3,3 of order 6 or the Petersen graph O3 of order 10. (4) is 4-regular if and only if is isomorphic to the Heawood graph of order 14. PROPOSITION 2.6 [26, Theorem 1.3] Let be a connected cubic arc-transitive graph of order 4p, p a prime. Then is one of the following: Q3, the 3-dimensional cube; D20, the dodecahedron; C28, the Coxeter graph; and G P(10, 3), the generalized Peterson graph, which is also the standard double cover of Petersen graph. 3 Page 4 of 8 Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:3 3. Proof of Theorem 1.1 Suppose to the contrary, is a hexavalent half-arc-transitive graph of order 2p2 for a prime p. As the smallest half-arc-transitive graph has order 27, we can assume that p ≥ 5. Let A = Aut () and let P be a Sylow p-subgroup of A.