Hamilton decompositions of one-ended Cayley graphs

TL;DR

This paper proves that certain infinite Cayley graphs, including all n-dimensional grids, can be decomposed into edge-disjoint Hamiltonian double-rays, advancing understanding of their Hamiltonian decompositions.

Contribution

It establishes the existence of Hamiltonian decompositions in one-ended, locally finite Cayley graphs with non-torsion generators, including all n-dimensional grids.

Findings

Any one-ended, locally finite Cayley graph with non-torsion generators admits a Hamiltonian double-ray decomposition.

The n-dimensional grid le  admits a decomposition into n edge-disjoint Hamiltonian double-rays.

The result applies to a broad class of Cayley graphs, including all le .

Abstract

We prove that any one-ended, locally finite Cayley graph with non-torsion generators admits a decomposition into edge-disjoint Hamiltonian (i.e. spanning) double-rays. In particular, the $n$-dimensional grid $\mathbb{Z}^n$ admits a decomposition into $n$ edge-disjoint Hamiltonian double-rays for all $n \in \mathbb{N}$.