Sharp threshold for hamiltonicity of random geometric graphs

TL;DR

This paper establishes a precise threshold for the emergence of Hamiltonian cycles in random geometric graphs under any $\, ext{l}_p$ norm, and provides an efficient algorithm to find such cycles above this threshold.

Contribution

It identifies the sharp threshold for Hamiltonicity in random geometric graphs for all $\, ext{l}_p$ norms and offers a linear time algorithm to find Hamiltonian cycles above this threshold.

Findings

Sharp threshold at $r(n)=\,\sqrt{\frac{\log n}{\alpha_p n}}$ for Hamiltonicity.

Constructive proof with a linear time algorithm for finding Hamiltonian cycles.

Algorithm works a.a.s. when $r(n)\ge\sqrt{\frac{\log n}{(\alpha_p -\epsilon)n}}$.

Abstract

We show for an arbitrary $\ell_p$ norm that the property that a random geometric graph $\mathcal G(n,r)$ contains a Hamiltonian cycle exhibits a sharp threshold at $r=r(n)=\sqrt{\frac{\log n}{\alpha_p n}}$, where $\alpha_p$ is the area of the unit disk in the $\ell_p$ norm. The proof is constructive and yields a linear time algorithm for finding a Hamiltonian cycle of $\RG$ a.a.s., provided $r=r(n)\ge\sqrt{\frac{\log n}{(\alpha_p -\epsilon)n}}$ for some fixed $\epsilon > 0$.