Infinitely many nonsolvable groups whose Cayley graphs are hamiltonian
Keywords:
Cayley graph, Hamiltonian cycle, Solvable group, Alternating groupAbstract
We show 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, we show that if $A_5$~is the alternating group on five letters, and $p$~is any prime, such that $p \equiv 1 \pmod{30}$, then every connected Cayley graph on the direct product $A_5 \times \integer _p$ has a hamiltonian cycle.
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References
R. Gould and R. Roth, Cayley digraphs and $(1, j, n)$-sequencings of the alternating groups $A_n$, Discrete Math. 66(1-2) (1987) 91-102.
K. Kutnar, D. Marušič, D. W. Morris, J. Morris, and P. Šparl, Hamiltonian cycles in Cayley graphs whose order has few prime factors, Ars Math. Contemp. 5(1) (2012) 27-71.
K. Kutnar, D. Marušič, D. W. Morris, J. Morris, and P. Šparl, Cayley graphs on $A_5$ are hamiltonian, unpublished appendix to [2], http://arxiv.org/src/1009.5795/anc/A5.pdf.
D. Witte and J. A. Gallian, A survey: Hamiltonian cycles in Cayley graphs, Discrete Math. 51(3) (1984) 293-304.
