On the chromatic number of random geometric graphs

AbstractGiven independent random points X1,...,Xn ∈ℝd with common probability distribution ν, and a positive distance r=r(n)>0, we construct a random geometric graph Gn with vertex set {1,..., n} where distinct i and j are adjacent when ‖Xi−Xj‖≤r. Here ‖·‖ may be any norm on ℝd, and ν may be any probability distribution on ℝd with a bounded density function. We consider the chromatic number χ(Gn) of Gn and its relation to the clique number ω(Gn) as n→∞. Both McDiarmid [11] and Penrose [15] considered the range of r when $$r \ll \left( {\tfrac{{\ln n}} {n}} \right)^{1/d}$$ and the range when $$r \gg \left( {\tfrac{{\ln n}} {n}} \right)^{1/d}$$ , and their results showed a dramatic difference between these two cases. Here we sharpen and extend the earlier results, and in particular we consider the ‘phase change’ range when $$r \sim \left( {\tfrac{{t\ln n}} {n}} \right)^{1/d}$$ with t>0 a fixed constant. Both [11] and [15] asked for the behaviour of the chromatic number in this range. We determine constants c(t) such that $$\tfrac{{\chi (G_n )}} {{nr^d }} \to c(t)$$ almost surely. Further, we find a “sharp threshold” (except for less interesting choices of the norm when the unit ball tiles d-space): there is a constant t0>0 such that if t≤t0 then $$\tfrac{{\chi (G_n )}} {{\omega (G_n )}}$$ tends to 1 almost surely, but if t>t0 then $$\tfrac{{\chi (G_n )}} {{\omega (G_n )}}$$ tends to a limit >1 almost surely.