On Cayley digraphs that do not have hamiltonian paths

TL;DR

This paper constructs an infinite family of connected Cayley digraphs without Hamiltonian paths, and identifies conditions on the group structure that guarantee the existence of such paths.

Contribution

It introduces a new family of Cayley digraphs lacking Hamiltonian paths and characterizes groups where all connected Cayley digraphs have Hamiltonian paths.

Findings

Infinite family of Cayley digraphs without Hamiltonian paths

Groups with |[G,G]| < 4 always have Hamiltonian paths in their Cayley digraphs

Counterexamples exist when |[G,G]| = 4 or 5

Abstract

We construct an infinite family of connected, 2-generated Cayley digraphs Cay(G;a,b) that do not have hamiltonian paths, such that the orders of the generators a and b are arbitrarily large. We also prove that if G is any finite group with |[G,G]| < 4, then every connected Cayley digraph on G has a hamiltonian path (but the conclusion does not always hold when |[G,G]| = 4 or 5).