  Following a problem posed by Lov\'asz in 1969, it is believed that every
connected vertex-transitive graph has a Hamilton path. This is shown here to be
true for cubic Cayley graphs arising from groups having a
$(2,s,3)$-presentation, that is, for groups $G=\la a,b| a^2=1, b^s=1, (ab)^3=1,
etc. \ra$ 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
vertices (and thus necessarily a Hamilton path) when $|G|$ is congruent to 0
modulo 4.
