Semiregular automorphisms of arc-transitive graphs with valency pqpq

It was conjectured (see [D. Marušič, On vertex symmetric digraphs, Discrete Math. 36 (1981) 69–81]) that every vertex-transitive digraph has a semiregular automorphism, that is, a nonidentity automorphism having all orbits of equal length. Despite several partial results supporting its content, the conjecture remains open. In this paper, it is shown that the conjecture holds whenever the graph is arc-transitive of valency pqpq, where pp and qq are primes (pp may equal qq), and such that its automorphism group has a nonabelian minimal normal subgroup with at least three orbits on the vertex set.