The existence of $m$-Haar graphical representations

TL;DR

This paper generalizes Haar graphical representations to m-partite graphs, providing a complete classification of finite groups that lack such representations, thus extending the understanding of group actions on graphs.

Contribution

It introduces m-Haar graphical representations as a natural extension of HGRs and classifies all finite groups that do not admit m-HGRs.

Findings

Complete classification of finite groups without m-HGRs.

Extension of HGR concept to m-partite graphs.

Addresses a more robust existence problem for GmSRs.

Abstract

Extending the well-studied concept of graphical regular representations to bipartite graphs, a Haar graphical representation (HGR) of a group $G$ is a bipartite graph whose automorphism group is isomorphic to $G$ and acts semiregularly with the orbits giving the bipartition. The question of which groups admit an HGR was inspired by a closely related question of Est\'elyi and Pisanski in 2016, as well as Babai's work in 1980 on poset representations, and has been recently solved by Morris and Spiga. In this paper, we introduce the $m$-Haar graphical representation ($m$-HGR) as a natural generalization of HGR to $m$-partite graphs for $m\geq2$, and explore the existence of $m$-HGRs for any fixed group. This inquiry represents a more robust version of the existence problem of G$m$SRs as addressed by Du, Feng and Spiga in 2020. Our main result is a complete classification of finite groups $G$ without $m$-HGRs.