Hamiltonicity of cubic Cayley graphs

Following a problem posed by Lov\'asz in 1969, it is believed that every finite connected vertex-transitive graph has a Hamilton path. This is shown here to be true for cubic Cayley graphs arising from finite groups having a $(2,s,3)$-presentation, that is, for groups $G=\la a,b\mid a^2=1, b^s=1, (ab)^3=1, \dots \ra$ generated by an involution $a$ and an element $b$ of order $s\geq3$ such that their product $ab$ has order $3$. More precisely, it is shown that the Cayley graph $X=Cay(G,\{a,b,b^{-1}\})$ has a Hamilton cycle when $|G|$ (and thus $s$) is congruent to $2$ modulo $4$, and has a long cycle missing only two adjacent vertices (and thus necessarily a Hamilton path) when $|G|$ is congruent to $0$ modulo $4$.