Hamiltonian cycles in cubic Cayley graphs: the 〈2,4k,3〉 case

It was proved by Glover and Marušič (J. Eur. Math. Soc. 9:775–787, 2007), that cubic Cayley graphs arising from groups G = 〈a, x | a2 = x = (ax)3 = 1, . . .〉 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.