Tetravalent half-arc-transitive graphs of order 2pq

Abstract

A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not arc set. Let p and q be primes. It is known that no tetravalent half-arc-transitive graphs of order 2p (2) exist and a tetravalent half-arc-transitive graph of order 4p must be non-Cayley; such a non-Cayley graph exists if and only if p-1 is divisible by 8 and it is unique for a given order. Based on the constructions of tetravalent half-arc-transitive graphs given by Marui (J. Comb. Theory B 73:41-76, 1998), in this paper the connected tetravalent half-arc-transitive graphs of order 2pq are classified for distinct odd primes p and q.