On the chromatic number of random geometric graphs

Given 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 (ln n n )1/d and the range when r (ln n n )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 ∼ (t ln n n )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 χ(Gn) nrd → c(t) almost surely. Furt...

• Type: Journal
• Year: 2011
• Where: COMBINATORICA
• Authors: Colin McDiarmid, Tobias Müller
• Keywords: Clique Number, COMBINATORICA 2011, Constants C, Geometric Graph, Optimization