On oriented m $m$‐semiregular representations of finite groups

Abstract

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 .