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 T₈, ₉ G (i, j Zₘ), we construct the m-Cayley digraph Γ= Cay (G, T₈, ₉: i, j Zₘ) with vertex set ₈䂷Gᵢ (where Gᵢ = \gᵢ | g G\) and arc set ₈, ₉\ (gᵢ, (tg) ⱼ) | t T₈, ₉, g G\. When T₈, ₈ = 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 m2.
Xu et al. (Sat,) studied this question.