The $m$-partite digraphical representations of valency 3 of finite groups generated by two elements

Abstract

Let $G$ be a finite group and $m$ be an integer. We employ the notation $g_i$ to represent elements $(g,i)$ in the Cartesian product $G\times {\mathbb{Z}}_m$, where ${\mathbb{Z}}_m$ denotes integers modulo $m$. For given sets $T_{i,j}\subseteq G$ ($i,j\in {\mathbb{Z}}_m$), we construct the $m$-$Cayley$ $digraph$ $\Gamma =\mathrm{Cay}(G,T_{i,j}:i,j\in {\mathbb{Z}}_m)$ with vertex set ${\bigcup }_{i\in {\mathbb{Z}}_m}{G}_i$ (where $G_i=\{g_i|g\in G\}$) and arc set ${\bigcup }_{i,j}\{(g_i,(tg{)}_j)|t\in T_{i,j},g\in G\}$. When $T_{i,i}=\varnothing$ for all $i\in {\mathbb{Z}}_m$, we call $\Gamma$ an \emph{$m$-partite Cayley digraph}. For $m$-partite Cayley digraphs, we observe that a $1$-partite Cayley digraph is necessarily an empty graph. Therefore, throughout this paper, we restrict our consideration to the case where $m\ge 2$. The digraph $\Sigma$ is regular if there exists a non-negative integer $k$ such that every vertex has out-valency and in-valency equal to $k$. All digraphs considered in this paper are regular. We say a group $G$ admits an \emph{$m$-partite digraphical representation} ($m$-PDR for short) if there exists a regular $m$-partite Cayley digraph $\Gamma$ with $\mathrm{Aut}(\Gamma )\cong G$. Based on Du et al.'s complete classification of unrestricted $m$-PDRs \cite{du4} (2022), we focus on the unresolved valency-specific cases. In this paper, we investigate $m$-PDRs of valency 3 for groups generated by at most two elements, and establish a complete classification of nontrivial finite simple groups admitting $m$-PDRs of valency 3 with $m\ge 2$.