Hamiltonicity of Cubic Cayley Graphs

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 $G=\la a,b| a^2=1, b^s=1, (ab)^3=1, etc. \ra$generatedbyaninvolution$a$andanelement$b$oforder$s\geq3$suchthattheirproduct$ab$ has order 3. More precisely, it is shown that the Cayley graph $X=Cay(G,\{a,b,b^{-1}\})$ has a Hamilton cycle when $\mathrm{\mid }G\mathrm{\mid }$ (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 $\mathrm{\mid }G\mathrm{\mid }$ is congruent to 0 modulo 4.