Pseudo-cartesian product and hamiltonian decompositions of Cayley graphs on abelian groups

Abstract

Alspach has conjectured that any 2 k-regular connected Cayley graph cay( A, S) on a finite abelian group A can be decomposed into k hamiltonian cycles. In this paper we generalize a result by Kotzig that the cartesian product of any two cycles can be decomposed into two hamiltonian cycles and show that any pseudo-cartesian product of two cycles can be decomposed into two hamiltonian cycles. By applying that result we first give an alternative proof for the main result in (Bermond et al., 1989), including the missing cases, and then we show that the conjecture is true for most 6-regular connected Cayley graphs on abelian groups of odd order and for some 6-regular connected Cayley graphs on abelian groups of even order.