Half-arc-transitive graphs of order 4p of valency twice a prime

Abstract

A graph is half-arc-transitive if its automorphism group acts transitively on vertices and edges, but not on arcs. Let p be a prime. Cheng and Oxley [On weakly symmetric graphs of order twice a prime, J. Combin. Theory B 42 (1987), 196−211] proved that there is no half-arc-transitive graph of order 2 p , and Alspach and Xu [1/2-transitive graphs of order 3 p , J. Algebraic Combin. 3 (1994), 347−355] classified half-arc-transitive graphs of order 3 p . In this paper we classify half-arc-transitive graphs of order 4 p of valency 2 q for each prime q ≥ 5. It is shown that such graphs exist if and only if p − 1 is divisible by 4 q . Moreover, for such p and q a unique half-arc-transitive graph of order 4 p and valency 2 q exists and this graph is a Cayley graph.