Hamiltonicity of the random geometric graph

TL;DR

This paper proves that in a random geometric graph, the emergence of the first Hamilton cycle coincides with the last vertex of degree less than two disappearing, extending classical random graph results to geometric settings.

Contribution

It establishes a precise relationship between Hamiltonicity and vertex degrees in random geometric graphs, answering an open question and generalizing known results.

Findings

First Hamilton cycle appears simultaneously with last degree < 2 vertex removal.

Provides exact limiting probability for Hamiltonicity in random geometric graphs.

Results extend to higher dimensions and different norms.

Abstract

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 $1<p\leq\infty$. The proof can also be adapted to show that, with probability tending to 1 as the number of points $n$ tends to $\infty$, there are cycles of all lengths between 3 and $n$ at the moment the graph loses its last vertex of degree less than two.