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 ( lnn n ) 1/d and the range when r ( lnn 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 lnn 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. 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 χ(Gn) ω(Gn) tends to 1 almost surely, but if t > t0 then χ(Gn) ω(Gn) tends to a limit > 1 almost surely.