# Hamiltonicity of cubic Cayley graphs

### Henry H. Glover

Ohio State University, Columbus, USA### Dragan Marusic

University of Ljubljana, Slovenia

## Abstract

Following a problem posed by Lovász 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 $G=⟨a,b∣a_{2}=1,b_{s}=1,(ab)_{3}=1,…⟩$ 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 $∣G∣$ (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 $4$.

## Cite this article

Henry H. Glover, Dragan Marusic, Hamiltonicity of cubic Cayley graphs. J. Eur. Math. Soc. 9 (2007), no. 4, pp. 775–787

DOI 10.4171/JEMS/96