Extending Cycles Locally to Hamilton Cycles

Julian Pott,Matthias Hamann,Florian Lehner

doi:10.37236/3960

Abstract

A Hamilton circle in an infinite graph is a homeomorphic copy of the unit circle $S^1$ that contains all vertices and all ends precisely once. We prove that every connected, locally connected, locally finite, claw-free graph has such a Hamilton circle, extending a result of Oberly and Sumner to infinite graphs. Furthermore, we show that such graphs are Hamilton-connected if and only if they are $3$-connected, extending a result of Asratian. Hamilton-connected means that between any two vertices there is a Hamilton arc, a homeomorphic copy of the unit interval $[0,1]$ that contains all vertices and all ends precisely once.

Highlights

The proofs of many classical sufficient conditions for the existence of a Hamilton cycle can be outlined as follows
A Hamilton circle in an infinite graph is a homeomorphic copy of the unit circle S1 that contains all vertices and all ends precisely once
We prove that every connected, locally connected, locally finite, claw-free graph has such a Hamilton circle, extending a result of Oberly and Sumner to infinite graphs

Summary

Introduction

The proofs of many classical sufficient conditions for the existence of a Hamilton cycle can be outlined as follows. [4, Proposition 3] A standard subspace X of |G| is a circle if and only if it is topologically connected and every vertex and end of G in X has degree 2. As C meets every finite cut in a positive even number of edges, every vertex and every end of C has even degree by Theorem 4 and C is topologically connected by Lemma 3 (i). Lemma 3 (ii) implies that the standard subspace C is arc-connected and the degree of each vertex and end is at least 2. This implies that the standard subspace C is a circle by Lemma 5 It is a Hamilton circle, as it contains every vertex and every end

Skip-and-glue extensions

Locally connected graphs

Further sufficient conditions for the existence of a Hamilton circle