2-generated Cayley digraphs on nilpotent groups have hamiltonian paths

Suppose G is a nilpotent, finite group. We show that if {a,b} is any 2-element generating set of G, then the corresponding Cayley digraph Cay(G;a,b) has a hamiltonian path. This implies there is a hamiltonian path in every connected Cayley graph on G that has valence at most 4.Comment: 7 pages, no figures; corrected a few typographical error

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

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 whose order has few prime factors

We prove that if Cay▫$(GS)$▫ is a connected Cayley graph with ▫$n$▫ vertices, and the prime factorization of ▫$n$▫ is very small, then Cay▫$(GS)$▫ has a hamiltonian cycle. More precisely, if ▫$p$▫, ▫$q$▫, and ▫$r$▫ are distinct primes, then ▫$n$▫ can be of the form kp with ▫$24 ne k < 32$▫, or of the form ▫$kpq$▫ with ▫$k le 5$▫, or of the form ▫$pqr$▫, or of the form ▫$kp^2$▫ with ▫$k le 4$▫, or of the form ▫$kp^3$▫ with ▫$k le 2$▫

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

Infinitely many nonsolvable groups whose Cayley graphs are hamiltonian

This note shows there are infinitely many finite groups G, such that every connected Cayley graph on G has a hamiltonian cycle, and G is not solvable. Specifically, for every prime p that is congruent to 1, modulo 30, we show there is a hamiltonian cycle in every connected Cayley graph on the direct product of the cyclic group of order p with the alternating group A_5 on five letters.Comment: 7 pages, plus a 22-page appendix of notes to aid the refere

Hamiltonian cycles in Cayley graphs whose order has few prime factors

We prove that if Cay▫$(GS)$▫ is a connected Cayley graph with ▫$n$▫ vertices, and the prime factorization of ▫$n$▫ is very small, then Cay▫$(GS)$▫ has a hamiltonian cycle. More precisely, if ▫$p$▫, ▫$q$▫, and ▫$r$▫ are distinct primes, then ▫$n$▫ can be of the form kp with ▫$24 ne k < 32$▫, or of the form ▫$kpq$▫ with ▫$k le 5$▫, or of the form ▫$pqr$▫, or of the form ▫$kp^2$▫ with ▫$k le 4$▫, or of the form ▫$kp^3$▫ with ▫$k le 2$▫