On oriented m-semiregular representations of finite groups about valency two

Abstract

Given a group G, an m-Cayley digraph Γ over G is a digraph that has a group of automorphisms isomorphic to G acting semiregularly on the vertex set with m orbits. We say that G admits an oriented m-semiregular representation (OmSR for short), if there exists a regular m-Cayley digraph Γ over G such that Γ is oriented and its automorphism group is isomorphic to G. In particular, O1SR is also named as ORR. Verret and Xia gave a classification of finite simple groups admitting an ORR of valency two in Verret and Xia (2022) [19]. Let m≥2 be an integer. In this paper, we show that all finite groups generated by at most two elements admit an OmSR of valency two except four groups of small orders. Consequently, a classification of finite simple groups admitting an OmSR of valency two is obtained.