Flows that are sums of hamiltonian cycles in Cayley graphs on abelian groups Original Research Article

If X is any connected Cayley graph on any finite abelian group, we determine precisely which flows on X can be written as a sum of hamiltonian cycles. (This answers a question of B. Alspach.) In particular, if the degree of X is at least 5, and X has an even number of vertices, then the flows that can be so written are precisely the even flows, that is, the flows f, such that image is divisible by 2. On the other hand, there are examples of degree 4 in which not all even flows can be written as a sum of hamiltonian cycles. Analogous results were already known, from work of B. Alspach, S.C. Locke, and D. Witte, for the case where X is cubic, or has an odd number of vertices.