Some local--global phenomena in locally finite graphs

TL;DR

This paper investigates how local properties of small-radius balls in infinite, locally finite graphs influence global Hamiltonian and connectivity properties, establishing conditions under which such graphs contain Hamiltonian curves.

Contribution

It introduces new conditions based on local ball properties that guarantee Hamiltonian and connectivity features in infinite graphs, extending classical finite graph results.

Findings

If every radius-2 ball is 2-connected and a degree sum condition holds, the graph has a Hamiltonian curve.

If every radius-1 ball satisfies Ore's condition, then all larger balls are Hamiltonian.

Local connectivity and degree conditions imply global Hamiltonian properties in infinite graphs.

Abstract

In this paper we present some results for a connected infinite graph $G$ with finite degrees where the properties of balls of small radii guarantee the existence of some Hamiltonian and connectivity properties of $G$. (For a vertex $w$ of a graph $G$ the ball of radius $r$ centered at $w$ is the subgraph of $G$ induced by the set $M_r(w)$ of vertices whose distance from $w$ does not exceed $r$). In particular, we prove that if every ball of radius 2 in $G$ is 2-connected and $G$ satisfies the condition $d_G(u)+d_G(v)\geq |M_2(w)|-1$ for each path $uwv$ in $G$, where $u$ and $v$ are non-adjacent vertices, then $G$ has a Hamiltonian curve, introduced by K\"undgen, Li and Thomassen (2017). Furthermore, we prove that if every ball of radius 1 in $G$ satisfies Ore's condition (1960) then all balls of any radius in $G$ are Hamiltonian.