Hamiltonian decompositions of Cayley graphs on abelian groups of even order

Alspach conjectured that any 2k-regular connected Cayley graph cay(A,S) on a finite abelian group A can be decomposed into k hamiltonian cycles. In 1992, the author proved that the conjecture holds if S={s1,s2, …, sk} is a minimal generating set of an abelian group A of odd order. Here we prove an analogous result for abelian group of even order: If A is a finite abelian group of even order at least 4 and S={s1,s2, …, sk} is a strongly minimal generating set (i.e., 2si∉〈S−{si}〉 for each 1⩽i⩽k) of A, then cay(A,S) can be decomposed into hamiltonian cycles.