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Sharp threshold for hamiltonicity of random geometric graphs

J. Diaz·D. Mitsche·X. Perez
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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 $r=r(n)=\sqrt{\frac{\log n}{α_p n}}$, where $α_p$ 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 $r=r(n)\ge\sqrt{\frac{\log n}{(α_p -ε)n}}$ for some fixed $ε> 0$.

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J. Diaz, D. Mitsche, X. Perez. 2006-07-07. Sharp threshold for hamiltonicity of random geometric graphs. https://arxiv.org/abs/cs/0607023

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