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

Let [Formula: see text] be a finite group and [Formula: see text] be an integer. We employ the notation [Formula: see text] to represent elements [Formula: see text] in the cartesian product [Formula: see text], where [Formula: see text] denotes integers modulo [Formula: see text]. For given sets [Formula: see text] ([Formula: see text]), we construct the [Formula: see text]-[Formula: see text] [Formula: see text] [Formula: see text] with vertex set [Formula: see text] (where [Formula: see text]) and arc set [Formula: see text]. When [Formula: see text] for all [Formula: see text], we call [Formula: see text] an [Formula: see text]-partite Cayley digraph. For [Formula: see text]-partite Cayley digraphs, we observe that a [Formula: see text]-partite Cayley digraph is necessarily an empty graph. Therefore, throughout this paper, we restrict our consideration to the case where [Formula: see text]. The digraph [Formula: see text] is [Formula: see text]-regular if there exists a nonnegative integer [Formula: see text] such that every vertex has out-valency and in-valency equal to [Formula: see text]. All digraphs considered in this paper are regular. We say a group [Formula: see text] admits an [Formula: see text]-partite digraphical representation ([Formula: see text]-PDR for short) if there exists a regular [Formula: see text]-partite Cayley digraph [Formula: see text] with [Formula: see text]. Based on Du et al.’s complete classification of unrestricted [Formula: see text]-PDRs [J.-L. Du, Y.-Q. Feng and P. Spiga, On [Formula: see text]-partite digraphical representations of finite groups, J. Comb. Theory Ser. A 189 (2022) 105606.], we focus on the unresolved valency-specific cases. In this paper, we investigate [Formula: see text]-PDRs of valency 3 for groups generated by at most two elements, and establish a complete classification of nontrivial finite simple groups admitting [Formula: see text]-PDRs of valency 3 with [Formula: see text].