Hamiltonicity of cubic Cayley graphs

Following a problem posed by Lovasz 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 = < a, b vertical bar a(2) = 1, b(s) = 1, ( ab)(3) = 1, ...> generated by an involution a and an element b of order s >= 3 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 vertical bar G vertical bar (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 vertical bar G vertical bar is congruent to 0 modulo 4.