A localization method in Hamiltonian graph theory

TL;DR

This paper develops a general approach to localize classical Hamiltonicity criteria in graphs, extending global conditions to local properties and applying these to finite and infinite graphs to identify Hamiltonian structures.

Contribution

The paper introduces a unified method for deriving localization theorems for Hamilton cycles, extending classical global criteria to local conditions in finite and infinite graphs.

Findings

Formulated local analogues of classical Hamiltonicity results.

Extended two localization theorems to infinite graphs.

Guaranteed Hamiltonian curves in certain infinite graphs.

Abstract

The classical global criteria for the existence of Hamilton cycles only apply to graphs with large edge density and small diameter. In a series of papers Asratian and Khachatryan developed local criteria for the existence of Hamilton cycles in finite connected graphs, which are analogues of the classical global criteria due to Dirac (1952), Ore (1960), Jung (1978), and Nash-Williams (1971). The idea was to show that the global concept of Hamiltonicity can, under rather general conditions, be captured by local phenomena, using the structure of balls of small radii. (The ball of radius $r$ centered at a vertex $u$ is a subgraph of $G$ induced by the set of vertices whose distances from $u$ do not exceed $r$.) Such results are called localization theorems and present a possibility to extend known classes of finite Hamiltonian graphs. In this paper we formulate a general approach for finding localization theorems and use this approach to formulate local analogues of well-known results of Bauer et al. (1989), Bondy (1980), H\"aggkvist and Nicoghossian (1981), and Moon and Moser (1963). Finally we extend two of our results to infinite locally finite graphs and show that they guarantee the existence of Hamiltonian curves, introduced by K\"undgen, Li and Thomassen (2017).