On oriented m m-semiregular representations of finite groups

A finite group admits an oriented regular representation if there exists a Cayley digraph of such that it has no digons and its automorphism group is isomorphic to . Let be a positive integer. In this paper, we extend the notion of oriented regular representations to oriented -semiregular representations using -Cayley digraphs. Given a finite group , an -Cayley digraph of is a digraph that has a group of automorphisms isomorphic to acting semiregularly on the vertex set with orbits. We say that a finite group admits an oriented -semiregular representation (OSR for short) if there exists an -Cayley digraph of such that it has no digons and is isomorphic to its automorphism group. Moreover, if is regular, that is, each vertex has the same in- and out-valency, we say is a regular oriented -semiregular representation (regular OSR for short) of . In this paper, we classify finite groups admitting a regular OSR or an OSR for each positive integer .