Hamiltonicity of the random geometric graph

Details Hamiltonicity of the random geometric graph Hamiltonicity of the random geometric graph

2009-05-30

arxiv.org/abs/0906.0071v2

Mathematics

Combinatorics

17 pages, 3 figures, improved the presentation, added concluding remarks and a section on generalizations to higher dimensions

Scientific paper

Let $X_1,..., X_n$ 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 $\infty$, 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, Koml\'os and Szemer\'edi and independently of Bollob\'as 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 Koml\'os and Szemer{\'e}di 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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