On oriented $m$-semiregular representations of finite groups

TL;DR

This paper extends the concept of oriented regular representations to oriented m-semiregular representations using m-Cayley digraphs, classifying finite groups that admit such representations for all positive integers m.

Contribution

It introduces and classifies finite groups admitting oriented m-semiregular representations via m-Cayley digraphs, generalizing previous notions for all positive integers m.

Findings

Classification of finite groups admitting oriented m-semiregular representations for each m

Extension of oriented regular representations to m-semiregular cases

Characterization of automorphism groups of m-Cayley digraphs

Abstract

A finite group $G$ admits an {\em 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 {\em $m$-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 {\em oriented $m$-semiregular representation} if there exists a regular $m$-Cayley digraph of $G$ such that it has no digons and $G$ is isomorphic to its automorphism group. In this paper, we classify finite groups admitting an oriented $m$-semiregular representation for each positive integer $m$.