Infinitely many nonsolvable groups whose Cayley graphs are hamiltonian

Yıl 2016, Cilt: 3 Sayı: 1, 13 - 30, 11.01.2016

Dave Witte Morris

https://doi.org/10.13069/jacodesmath.66457

Öz

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.

Anahtar Kelimeler

Cayley graph, Hamiltonian cycle, Solvable group, Alternating group

Kaynakça

• R. Gould, R. Roth, Cayley digraphs and (1, j, n)-sequencings of the alternating groups An, Discrete Math. 66(1-2) (1987) 91–102.
• K. Kutnar, D. Marušič, D. W. Morris, J. Morris, 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• K. Kutnar, D. Marušič, D. W. Morris, J. Morris, P. Šparl, Cayley graphs on A5are hamiltonian, unpublished appendix to [2], http://arxiv.org/src/1009.5795/anc/A5.pdf.
• D. Witte, J. A. Gallian, A survey: Hamiltonian cycles in Cayley graphs, Discrete Math. 51(3) (1984) 293–304.