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

TL;DR

This paper extends the concept of oriented regular representations to m-partite oriented graphs, classifies which finite groups admit such structures, and introduces the notions of m-POSR and m-HOR.

Contribution

It introduces the concepts of m-partite oriented semiregular and Haar representations and provides a complete classification of finite groups lacking these representations for m≥2.

Findings

Classified finite groups without m-HORs or m-POSRs for m≥2.

Extended the concept of ORR to m-partite oriented graphs.

Connected the structures to m-Haar graphs and their automorphism groups.

Abstract

The study of ORR was inspired by L\'{a}zsl\'{o} Babai in 1980 when he asked a question: Which [finite] groups admit an oriented graph as a DRR? And it has been solved by Joy Morris and Pablo Spiga through a series of papers in 2018. In this paper, we will extend the concept of ORR to $m$-partite oriented graphs for $m\geq 2$. We say that a finite group $G$ admits an \emph{$m$-partite oriented semiregular representation} ($m$-POSR) if there exists an $m$-partite oriented graph $\G$ such that its automorphism group is isomorphic to $G$ and acts semiregularly with the $m$ orbits giving the partition. Moreover, if $\G$ is regular, that is, each vertex has the same in- and out-valency, it can be viewed as the oriented version of an $m$-Haar graph of $G$ and we call $\G$ is an \emph{$m$-Haar oriented representation} ($m$-HOR) of $G$. Our main result is a complete classification of finite groups $G$ without $m$-HORs or $m$-POSRs for $m\geq 2$.