Hamiltonicity of the random geometric graph

2009/05/30 by Krivelevich, Michael, Muller, Tobias
#Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.0906.0071

Abstract

Let X₁,..., X_n be independent, uniformly random points from [0,1]². We prove that if we add edges between these points one by one by order of increasing edge length then, with probability tending to 1 as the number of points n tends to ∞, the resulting graph gets its first Hamilton cycle at exactly the same time it loses its last vertex of degree less than two. This answers an open question of Penrose and provides an analogue for the random geometric graph of a celebrated result of Ajtai, Komlós and Szemerédi and independently of Bollobás on the usual random graph. We are also able to deduce very precise information on the limiting probability that the random geometric graph is Hamiltonian analogous to a result of Komlós and Szemerédi on the usual random graph. The proof generalizes to uniform random points on the d-dimensional hypercube where the edge-lengths are measured using the l_p-norm for some 1

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