Hamiltonian cycles in Cayley graphs of imprimitive complex reflection groups

Generalizing a result of Conway, Sloane, and Wilkes for real reflection groups, we show the Cayley graph of an imprimitive complex reflection group with respect to standard generating reflections has a Hamiltonian cycle. This is consistent with the long-standing conjecture that for every finite group, G, and every set of generators, S, of G the undirected Cayley graph of G with respect to S has a Hamiltonian cycle.Comment: 15 pages, 4 figures; minor revisions according to referee comments, to appear in Discrete Mathematic

On Cayley digraphs that do not have hamiltonian paths

We construct an infinite family of connected, 2-generated Cayley digraphs Cay(G;a,b) that do not have hamiltonian paths, such that the orders of the generators a and b are arbitrarily large. We also prove that if G is any finite group with |[G,G]| < 4, then every connected Cayley digraph on G has a hamiltonian path (but the conclusion does not always hold when |[G,G]| = 4 or 5).Comment: 10 pages, plus 14-page appendix of notes to aid the refere