Hamilton decompositions of 6-regular Cayley graphs on even Abelian groups with involution-free connections sets

Abstract

Alspach conjectured that every connected Cayley graph on a finite Abelian group A is Hamilton-decomposable. Liu has shown that for |A| even, if S={s1,…,sk}⊂A is an inverse-free strongly minimal generating set of A, then the Cayley graph Cay(A;S⋆), is decomposable into k Hamilton cycles, where S⋆ denotes the inverse-closure of S. Extending these techniques and restricting to the 6-regular case, this article relaxes the constraint of strong minimality on S to require only that S be strongly a-minimal, for some a∈S and the index of 〈a〉 be at least four. Strong a-minimality means that 2s∉〈a〉 for all s∈S∖{a,−a}. Some infinite families of open cases for the 6-regular Cayley graphs on even order Abelian groups are resolved. In particular, if |s1|≥|s2|>2|s3|, then Cay(A;{s1,s2,s3}⋆) is Hamilton-decomposable.