  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).
