# Infinitely many nonsolvable groups whose Cayley graphs are hamiltonian

Keywords:
Cayley graph, Hamiltonian cycle, Solvable group, Alternating group

### Abstract

We show there are infinitely many finite groups~$G$, such that every connected Cayley graph on~$G$ has a hamiltonian cycle, and $G$ is not solvable. Specifically, we show that if $A_5$~is the alternating group on five letters, and $p$~is any prime, such that $p \equiv 1 \pmod{30}$, then every connected Cayley graph on the direct product $A_5 \times \integer _p$ has a hamiltonian cycle.