A classification of tightly attached half-arc-transitive graphs of valency 4

A graph is said to be half-arc-transitive if its automorphism group acts transitively on the set of its vertices and edges but not on the set of its arcs. With each half-arc-transitive graph of valency 4 a collection of the so called alternating cycles is associated, all of which have the same even length. Half of this length is called the radius of the graph in question. Moreover, any two adjacent alternating cycles have the same number of common vertices. If this number, the so called attachment number, coincides with the radius, we say that the graph is tightly attached. Marušič gave a classification of tightly attached half-arc-transitive graphs of valency 4 with odd radius. In this paper the even radius tightly attached graphs of valency 4 are classified, thus completing the classification of all tightly attached half-arc-transitive graphs of valency 4. 1 Introductory remarks Throughout this paper graphs are assumed to be finite and, unless stated otherwise, simple, connected and undirected. For group-theoretic concepts not defined here we refer the reader to [3, 8, 31], and for graph-theoretic terms not defined here we refer the reader to [4]. In this paper we let Z Z n denote the ring of residue classes modulo n and we let Z Z * n denote the set of invertible elements of Z Z n. At times it will be convenient to view elements of Z Z n as integers, for instance if ρ is an element of some group with ρ n = 1 and if r ∈ Z Z n , we let ρ r represent ρ k for any k in the equivalence class r. This should cause no confusion. For basic notation and other conventions see Section 2. Let X be a graph. We let V (X), E(X) and A(X) denote the set of vertices, edges and arcs of X, respectively. The graph X is said to be vertex-transitive, edge-transitive and arc-transitive provided that its automorphism group AutX acts transitively on the set of its vertices, edges and arcs, respectively. Moreover, X is said to be half-arc-transitive if it is vertex-and edge-but not arc-transitive. More generally, by a half-arc-transitive action of a subgroup G ≤ AutX on X we mean a vertex-and edge-but not arc-transitive action of G on X. In this case we say that X is G-half-arc-transitive. As demonstrated in [29, 7.53, p. 59] by Tutte, the …