Sharp Threshold for Hamiltonicity of Random Geometric Graphs

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 $\mathcal{G}(n,r)$ asymptotically almost surely, provided $r=r(n)\ge\sqrt{\frac{\log n}{(\alpha_p -\epsilon)n}}$ for some fixed $\epsilon>0$.