Which Haar graphs are Cayley graphs? (Q311497)

Summary: 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 \(D_{5}\).