Hamilton cycles in random geometric graphs

TL;DR

This paper establishes that in certain random geometric graph models, the emergence of Hamilton cycles coincides with 2-connectivity, answering longstanding questions and identifying connectivity thresholds for Hamiltonicity.

Contribution

It proves that in the Gilbert model, Hamiltonicity occurs precisely at 2-connectivity, and in the k-nearest neighbor model, a fixed connectivity level guarantees Hamilton cycles.

Findings

Hamilton cycles appear exactly when the graph becomes 2-connected in the Gilbert model.

A constant connectivity level ensures Hamiltonicity in the k-nearest neighbor model.

Answers to Penrose's question about Hamiltonian thresholds in random geometric graphs.

Abstract

We prove that, in the Gilbert model for a random geometric graph, almost every graph becomes Hamiltonian exactly when it first becomes 2-connected. This answers a question of Penrose. We also show that in the k-nearest neighbor model, there is a constant \kappa\ such that almost every \kappa-connected graph has a Hamilton cycle.