István Estélyi: Which Haar graphs are Cayley graphs?

Povzetek. For a finite group G and a subset S of G, a dipole with |S| parallel arcs labeled with elements of S, considered  as a voltage graph, admits a regular covering graph, denoted by H(G,S), which is a bipartite regular graph, called a Haar graph. 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 talk I am going to address the problem of classifying finite non-abelian groups G with the property that every Haar graph H(G,S) is a Cayley graph. We will deduce an equivalent condition for H(G,S)to be a Cayley graph of a group containing G in terms of G, S and Aut(G). We will see that the dihedral groups, which are solutions to the above problem, are Z₂²,D₆,D₈ and D₁₀.