$m$-partite oriented semiregular representation of valency 3 for finite groups

TL;DR

This paper classifies finite groups generated by at most two elements that admit specific oriented and multipartite Cayley digraph representations with automorphism group isomorphic to the group, extending previous classifications.

Contribution

It introduces the classification of $m$-partite oriented semiregular representations and provides a complete classification for groups admitting $m$-PDR of valency 3.

Findings

Classified finite groups with $m$-POSR for valency 3.

Established that $m$-POSR implies $m$-PDR, but not vice versa.

Extended previous classifications to multipartite digraphs.

Abstract

Let $G$ be a finite group and $m \geq 2$ a positive integer. We say that $G$ admits an \emph{oriented $m$-semiregular representation} (abbreviated as OmSR) if there exists a $m$-Cayley digraph $\Gamma$ over $G$ such that $\Gamma$ is oriented and $\mathrm{Aut}(\Gamma) \cong G$. In \cite{xu1}, we classified finite groups generated by at most two elements that admit an OmSR of valency 3 for $m \geq 2$ and $G \ncong \mathbb{Z}_1$. In this article, we consider $m$-partite digraphs.We say a finite group $G$ admits an \emph{$m$-partite oriented semiregular representation} ($m$-partite digraphical representation), abbreviated as \emph{$m$-POSR} (\emph{$m$-PDR}), if there exists an \emph{oriented} $m$-partite Cayley digraph (\emph{$m$-partite Cayley digraph}) $\Gamma$ with $\mathrm{Aut}(\Gamma) \cong G$. In this paper, we classify finite groups generated by at most two elements that admit $m$-POSR. Since if $G$ admits an $m$-POSR, then $G$ must also admit an $m$-PDR (while the converse does not hold), as a natural consequence, we also provide a complete classification for groups $G=\langle x,y\rangle$ that admit $m$-PDR of valency 3. This complements the results in \cite{xu2}.