Hamiltonian decompositions of 4‐regular Cayley graphs of infinite abelian groups

A well‐known conjecture of Alspach says that every2 k$2k$ ‐regular Cayley graph of a finite abelian group can be decomposed into Hamiltonian cycles. We consider an analogous question for infinite abelian groups. In this setting one natural analogue of a Hamiltonian cycle is a spanning double‐ray. However, a naive generalisation of Alspach's conjecture fails to hold in this setting due to the existence of2 k$2k$ ‐regular Cayley graphs with finite cutsF$F$ , where∣ F ∣$| F| $ andk$k$ differ in parity, which necessarily preclude the existence of a decomposition into spanning double‐rays. We show that every 4‐regular Cayley graph of an infinite abelian group all of whose finite cuts are even can be decomposed into spanning double‐rays, and so characterise when such decompositions exist. We also characterise when such graphs can be decomposed either into Hamiltonian circles, a more topological generalisation of a Hamiltonian cycle in infinite graphs, or into a Hamiltonian circle and a spanning double‐ray.