  The polycirculant conjecture asserts that every vertex-transitive digraph has
a semiregular automorphism, that is, a nontrivial automorphism whose cycles all
have the same length. In this paper we investigate the existence of semiregular
automorphisms of edge-transitive graphs. In particular, we show that any
regular edge-transitive graph of valency three or four has a semiregular
automorphism.
