Existence of non-Cayley Haar graphs

A Cayley graph of a group H is a finite simple graph Gamma such that its automorphism group Aut(Gamma) contains a subgroup isomorphic to H acting regularly on V(Gamma), while a Haar graph of H is a finite simple bipartite graph Sigma such that Aut(Sigma) contains a subgroup isomorphic to H acting semiregularly on V(Sigma) and the H-orbits are equal to the partite sets of Sigma. It is well-known that every Haar graph of finite abelian groups is a Cayley graph. In this paper, we prove that every finite non-abelian group admits a non-Cayley Haar graph except the dihedral groups D-6, D-8, D-10, the quaternion group Q(8) and the group Q(8) x Z(2). This answers an open problem proposed by Estelyi and Pisanski in 2016. (C) 2020 Elsevier Ltd. All rights reserved.