Hamilton decompositions of certain 6-regular Cayley graphs on Abelian groups with a cyclic subgroup of index two

Alspach conjectured that every connected Cayley graph on a finite Abelian group A is Hamiltondecomposable. 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.