On the chromatic number of random geometric graphs

Abstract

Given independent random points X 1,...,X n ∈ℝd with common probability distribution ν, and a positive distance r=r(n)>0, we construct a random geometric graph G n with vertex set {1,..., n} where distinct i and j are adjacent when ‖X i X j ‖≤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(G n ) of G n and its relation to the clique number ω(G n ) 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 t 0>0 such that if tt 0 then \(\tfrac{{\chi (G_n )}} {{\omega (G_n )}}\) tends to 1 almost surely, but if t>t 0 then \(\tfrac{{\chi (G_n )}} {{\omega (G_n )}}\) tends to a limit >1 almost surely.

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Correspondence to Colin Mcdiarmid.

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The research in this paper was conducted while this author was a research student at the University of Oxford. He was partially supported by Bekker-la-Bastide fonds, Hendrik Muller’s Vaderlandsch fonds, EPSRC, Oxford University Department of Statistics and Prins Bernhard Cultuurfonds.

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Mcdiarmid, C., Müller, T. On the chromatic number of random geometric graphs. Combinatorica 31, 423–488 (2011). https://doi.org/10.1007/s00493-011-2403-3

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Mathematics Subject Classification (2000)

  • 05C80
  • 60D05
  • 05C15