Hamiltonicity in locally finite graphs: two extensions and a counterexample

TL;DR

This paper extends classical Hamiltonicity results to locally finite graphs, providing new conditions for Hamilton circles, a uniqueness property, and constructing an example of a regular infinite graph with a unique Hamilton circle.

Contribution

It introduces two extensions of Hamiltonicity conditions to locally finite graphs and constructs a regular infinite graph with a unique Hamilton circle, addressing open questions.

Findings

A sufficient condition for the square of a locally finite graph to contain a Hamilton circle.

An alternative proof that certain 2-connected graphs without specific minors are Hamiltonian.

Construction of a regular infinite graph with a unique Hamilton circle.

Abstract

We state a sufficient condition for the square of a locally finite graph to contain a Hamilton circle, extending a result of Harary and Schwenk about finite graphs. We also give an alternative proof of an extension to locally finite graphs of the result of Chartrand and Harary that a finite graph not containing $K^4$ or $K_{2,3}$ as a minor is Hamiltonian if and only if it is $2$-connected. We show furthermore that, if a Hamilton circle exists in such a graph, then it is unique and spanned by the $2$-contractible edges. The third result of this paper is a construction of a graph which answers positively the question of Mohar whether regular infinite graphs with a unique Hamilton circle exist.