Joy Morris — Page 4

arXiv2004

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

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.

math.CO↗

arXiv1997

Automorphism groups with cyclic commutator subgroup and Hamilton cycles

It has been shown that there is a Hamilton cycle in every connected Cayley graph on each group G whose commutator subgroup is cyclic of prime-power order. This paper considers connected, vertex-transitive graphs X of order at least 3 where the automorphism group of X contains a transitive subgroup G whose commutator subgroup is cyclic of prime-power order. We show that of these graphs, only the Petersen graph is not hamiltonian.

math.CO↗