95 research outputs found
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
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
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
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