Semiregular automorphisms of semisymmetric graphs of twice prime valency

This paper is closely related to research on the Polycirculant Conjecture, which asserted that every vertex-transitive digraph with at least two vertices admits a semiregular automorphism, and more generally, that every transitive 2-closed permutation group of degree at least two contains a fixed-point-free element of prime order. Very recently, however, Freedman and Lee [8] disproved the Polycirculant Conjecture by constructing an infinite family of counterexamples. Nevertheless, determining under what conditions the conjecture's conclusion remains valid is a meaningful question. An interesting related problem asks whether analogous semiregular automorphisms must exist for regular graphs admitting an edge-transitive but not vertex-transitive group of automorphisms G . Such graphs are called G -semisymmetric; they are bipartite, with the two bipartition sets coinciding with the two orbits of G . In this paper, we prove that every connected G -semisymmetric graph of valency 2 p , where p is an odd prime, admits a semiregular automorphism preserving the bipartition, provided that the cardinality of each bipartition set is squarefree.