Hamiltonian cycles in cubic Cayley graphs: The $\langle 2,4k,3\rangle $ case

Glover, Henry H. ; Kutnar, Klavdija ; Marušič, Dragan

Journal of Algebraic Combinatorics, Tome 30 (2009) no. 4, pp. 447-475.

Résumé

Summary: It was proved by Glover and Marušič (J. Eur. Math. Soc. 9:775-787, 2007), that cubic Cayley graphs arising from groups $G=\langle a, x\mid a ^{2}= x ^{ s }=( ax) ^{3}=1,\cdots \rangle $having a $(2, s,3)$-presentation, that is, from groups generated by an involution $a$ and an element $x$ of order $s$ such that their product $ax$ has order 3, have a Hamiltonian cycle when $| G| (and thus also s)$ is congruent to 2 modulo 4, and have a Hamiltonian path when $| G|$ is congruent to 0 modulo 4. In this article the existence of a Hamiltonian cycle is proved when apart from $| G|$ also $s$ is congruent to 0 modulo 4, thus leaving $| G|$ congruent to 0 modulo 4 with $s$ either odd or congruent to 2 modulo 4 as the only remaining cases to be dealt with in order to establish existence of Hamiltonian cycles for this particular class of cubic Cayley graphs.

Détail

Keywords: keywords Hamiltonian cycle, Cayley graph, cubic arc-transitive graph, consistent cycle, cyclic edge connectivity