Disjoint Hamilton cycles in the random geometric graph

TL;DR

This paper proves a conjecture about the order in which Hamiltonian cycles and connectivity appear in the random geometric graph process, showing they occur simultaneously with high probability.

Contribution

It establishes that the first edge to achieve k-connectivity also creates multiple disjoint Hamilton cycles in the process, extending previous conjectures.

Findings

First edge making the graph Hamiltonian coincides with 2-connectivity.

For arbitrary k, the first k-connected graph contains k/2 disjoint Hamilton cycles or a similar structure.

Results hold asymptotically almost surely in the random geometric graph process.

Abstract

We prove a conjecture of Penrose about the standard random geometric graph process, in which n vertices are placed at random on the unit square and edges are sequentially added in increasing order of lengths taken in the l_p norm. We show that the first edge that makes the random geometric graph Hamiltonian is a.a.s. exactly the same one that gives 2-connectivity. We also extend this result to arbitrary connectivity, by proving that the first edge in the process that creates a k-connected graph coincides a.a.s. with the first edge that causes the graph to contain k/2 pairwise edge-disjoint Hamilton cycles (for even k), or (k-1)/2 Hamilton cycles plus one perfect matching, all of them pairwise edge-disjoint (for odd k).