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

Let G be a finite group and m be an integer. We employ the notation gᵢ to represent elements $(g,i)$ in the Cartesian product G × Zₘ, where Zₘ denotes integers modulo m. For given sets Ti,j ⊆ G (i,j ∈ Zₘ), we construct the m-$Cayley$ $digraph$ Γ= Cay(G, Ti,j: i,j ∈ Zₘ) with vertex set iₘGᵢ (where Gᵢ = ᵢ | g ∈ G\) and arc set i,j\(gᵢ, (tg)ⱼ) | t ∈ Ti,j, g ∈ G\. When Ti,i = ∅ for all i ∈ Zₘ, we call $Γ$ an 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 ≥ 2. The digraph $Σ$ 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 m-partite digraphical representation (m-PDR for short) if there exists a regular m-partite Cayley digraph $Γ$ with Aut(Γ) G. Based on Du et al.'s complete classification of unrestricted m-PDRs {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≥2.