Existence of non-Cayley Haar graphs

Abstract

A Cayley graph of a group H is a finite simple graph Γ such that its automorphism group Aut(Γ) contains a subgroup isomorphic to H acting regularly on V(Γ), while a Haar graph of H is a finite simple bipartite graph Σ such that Aut(Σ) contains a subgroup isomorphic to H acting semiregularly on V(Σ) and the H-orbits are equal to the partite sets of Σ. 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 D6, D8, D10, the quaternion group Q8 and the group Q8×Z2. This answers an open problem proposed by Estélyi and Pisanski in 2016.