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

Details Flows that are sums of hamiltonian cycles in Cayley graphs on abelian groups Flows that are sums of hamiltonian cycles in Cayley graphs on abelian groups

2003-09-02

arxiv.org/abs/math/0309050v2

Mathematics

Combinatorics

Latex2e file, 68 pages, minor errors corrected and title slightly changed

Scientific paper

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 Brian 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 the sum of the edge-flows of f 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 Alspach, Locke, and Witte, for the case where X is cubic, or has an odd number of vertices.

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