Which Haar graphs are Cayley graphs?

TL;DR

This paper investigates which Haar graphs derived from finite groups are also Cayley graphs, providing a classification for non-abelian groups and identifying specific groups like dihedral groups that satisfy this property.

Contribution

The paper classifies finite non-abelian groups for which all Haar graphs are Cayley graphs and establishes conditions involving automorphisms.

Findings

Dihedral groups are solutions: Z2^2, D3, D4, D5.

An equivalent condition for Haar graphs to be Cayley graphs is derived.

Provides a classification of non-abelian groups with this property.

Abstract

For a finite group $G$ and subset $S$ of $G,$ the Haar graph $H(G,S)$ is a bipartite regular graph, defined as a regular $G$-cover of a dipole with $|S|$ parallel arcs labelled by elements of $S$. If $G$ is an abelian group, then $H(G,S)$ is well-known to be a Cayley graph; however, there are examples of non-abelian groups $G$ and subsets $S$ when this is not the case. In this paper we address the problem of classifying finite non-abelian groups $G$ with the property that every Haar graph $H(G,S)$ is a Cayley graph. An equivalent condition for $H(G,S)$ to be a Cayley graph of a group containing $G$ is derived in terms of $G, S$ and $\mathrm{Aut }G$. It is also shown that the dihedral groups, which are solutions to the above problem, are $\mathbb{Z}_2^2,D_3,D_4$ and $D_{5}$.