## Abstract

Given independent random points *X*
_{1},...,*X*
_{
n
} ∈ℝ^{d} 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 ℝ^{d}, and *ν* may be any probability distribution on ℝ^{d} 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 *t*≤*t*
_{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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## Additional information

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