Infinitely many nonsolvable groups whose Cayley graphs are hamiltonian

TL;DR

This paper proves the existence of infinitely many non-solvable finite groups where every connected Cayley graph admits a Hamiltonian cycle, expanding understanding of Hamiltonian properties in group theory.

Contribution

It introduces a new class of non-solvable groups with universal Hamiltonian cycles in their Cayley graphs, specifically involving direct products with A_5.

Findings

For primes p ≡ 1 mod 30, all connected Cayley graphs on Z_p × A_5 have Hamiltonian cycles.

Infinitely many such non-solvable groups exist with this property.

The result applies to a broad class of groups constructed from cyclic groups and A_5.

Abstract

This note shows 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, for every prime p that is congruent to 1, modulo 30, we show there is a hamiltonian cycle in every connected Cayley graph on the direct product of the cyclic group of order p with the alternating group A_5 on five letters.