
István Estélyi

Tomaž Pisanski
Keywords:
Haar graph, Cayley graph, Dihedral group, Generalized dihedral group
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 wellknown to be a Cayley graph; however, there are examples of nonabelian groups $G$ and subsets $S$ when this is not the case. In this paper we address the problem of classifying finite nonabelian 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 $D_{5}$.