A classification of the m-graphical regular representation of finite groups

TL;DR

This paper classifies finite groups that admit m-graphical regular representations, extending classical notions and providing explicit descriptions for various values of m, including new existence results for groups with multiple orbits.

Contribution

It generalizes the classification of graphical regular representations to m-graphs, offering explicit descriptions and new existence results for all positive integers m.

Findings

Every non-identity finite group admits an m-GRR for all m>4.

Classified finite groups admitting m-GRR and m-DRR for arbitrary m.

Extended classical classifications to broader m-graphical regular representations.

Abstract

In this paper we extend the classical notion of digraphical and graphical regular representation of a group and we classify, by means of an explicit description, the finite groups satisfying this generalization. A graph or digraph is called regular if each vertex has the same valency, or, the same out-valency and the same in-valency, respectively. An m-(di)graphical regular representation (respectively, m-GRR and m-DRR, for short) of a group G is a regular (di)graph whose automorphism group is isomorphic to G and acts semiregularly on the vertex set with m orbits. When m=1, this definition agrees with the classical notion of GRR and DRR. Finite groups admitting a 1-DRR were classified by Babai in 1980, and the analogue classification of finite groups admitting a 1-GRR was completed by Godsil in 1981. Pivoting on these two results in this paper we classify finite groups admitting an m-GRR or an m-DRR, for arbitrary positive integers m. For instance, we prove that every non-identity finite group admits an m-GRR, for every m>4.