Arc-transitive graphs of valency 8 have a semiregular automorphism

Abstract

One version of the polycirculant conjecture states that every vertex-transitive graph has a non-identity semiregular automorphism that is, a non-identity automorphism whose cycles all have the same length.  We give a proof of the conjecture in the arc-transitive case for graphs of valency 8, which was the smallest open valency.