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

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