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

Abstract

Given a group $G$, an {\em $m$-Cayley digraph $\G$ 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 {\em oriented $m$-semiregular representation} (O$m$SR for short), if there exists a regular $m$-Cayley digraph $\G$ over $G$ such that $\G$ is oriented and its automorphism group is isomorphic to $G$. In particular, O$1$SR is also named as ORR. Verret and Xia gave a classification of finite simple groups admitting an ORR of valency two in [Ars Math. Contemp. 22 (2022), \#P1.07]. Let $m\ge 2$ be an integer. In this paper, we show that all finite groups generated by at most two elements admit an O$m$SR of valency two except four groups of small orders. Consequently, a classification of finite simple groups admitting an O$m$SR of valency two is obtained.