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

TL;DR

The paper addresses the problem of classifying finite groups $G$ generated by at most two elements that admit an oriented $m$-partite semiregular representation ($m$-POSR) of valency $3$, equivalently an oriented $m$-Cayley digraph with $ ext{Aut}( ext{Cay}(G,T_{i,j}))\cong G$. It develops the $m$-Cayley framework, analyzes automorphism groups via stabilizer arguments, and provides explicit constructions of connection sets $T_{i,j}$, supplemented by computational checks, to prove the main results for $G=\<x ight\rangle$ and $G=\<x,y ight angle$ across $m\ge 2$. The key contributions include a complete classification of two-generator groups admitting a $m$-POSR of valency $3$ (with identified exceptions such as certain small groups) and the corollary classification for $m$-PDR, along with constructive families for $m\ge 3$. This work extends the theory of graphical regular representations to oriented $m$-partite settings, enabling precise realizations of automorphism groups as $m$-Cayley digraphs and advancing understanding of group-graph correspondences.

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 $Γ$ over $G$ such that $Γ$ is oriented and $\mathrm{Aut}(Γ) \cong G$. 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}) $Γ$ with $\mathrm{Aut}(Γ) \cong G$. 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}.

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Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Proposition 1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3