Cayley graphs of order kp are hamiltonian for k < 48

We provide a computer-assisted proof that if G is any finite group of order kp, where k < 48 and p is prime, then every connected Cayley graph on G is hamiltonian (unless kp = 2). As part of the proof, it is verified that every connected Cayley graph of order less than 48 is either hamiltonian connected or hamiltonian laceable (or has valence less than three).Comment: 16 pages. GAP source code is available in the ancillary file

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

Cayley graphs of order 27p are hamiltonian

Suppose G is a finite group, such that |G| = 27p, where p is prime. We show that if S is any generating set of G, then there is a hamiltonian cycle in the corresponding Cayley graph Cay(G;S).Comment: 13 pages, no figures; minor revisions, including suggestions from a referee; this version is to appear in the International Journal of Combinatoric