Hamiltonicity in generalized quasi-dihedral groups

TL;DR

This paper proves that all connected Cayley graphs of infinite dihedral and certain generalized quasi-dihedral groups contain Hamiltonian double rays, extending known finite group results to infinite cases.

Contribution

It extends the known Hamiltonian path results from finite dihedral groups to infinite and more general quasi-dihedral groups, including the existence of Hamiltonian double rays.

Findings

Connected Cayley graphs of infinite dihedral groups have Hamiltonian double rays.

Extension of Hamiltonian path results to all two-ended generalized quasi-dihedral groups.

Demonstrates Hamiltonian properties in infinite group Cayley graphs.

Abstract

Witte Morris showed in [21] that every connected Cayley graph of a finite (generalized) dihedral group has a Hamiltonian path. The infinite dihedral group is defined as the free product with amalgamation $\mathbb Z_2 \ast \mathbb Z_2$. We show that every connected Cayley graph of the infinite dihedral group has both a Hamiltonian double ray, and extend this result to all two-ended generalized quasi-dihedral groups.