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

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 = { s 1 , … , s k } ⊂ 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 2 s ∉ 〈 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 | s 1 | ≥ | s 2 | > 2 | s 3 | , then Cay ( A ; { s 1 , s 2 , s 3 } ⋆ ) is Hamilton-decomposable.