Quasi-abelian cayley graphs and parsons graphs

LetGbe a finite group, andSa subset ofGwith 1∉SandS-1=S.IfSis a union of conjugacy classes ofG,then the Cayley graphX=Cay(G, S)onGwith respect toSis calledquasi-abelian.In this paper we prove that any connected quasi-abelian Cayley graph is hamiltonian (Theorem 2). As a consequence, we prove that Zaksʹ three conjectures on Parsons graphs are true, with one exception.