Disjoint Hamilton cycles in the random geometric graph
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 lp 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).
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