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 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.
Similar content being viewed by others
References
B. Bollobás, Random graphs, volume 73 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, second edition, 2001.
V. Chvátal: Linear Programming, W. H. Freedman and Company, New York, 1983.
B. N. Clark, C. J. Colbourn and D. S. Johnson: Unit disk graphs. Discrete Math. 86(1–3) (1990), 165–177.
J. Glaz, J. Naus and S. Wallenstein: Scan Statistics, Springer, New York, 2001.
A. Gräf, M. Stumpf and G. Weißenfels: On coloring unit disk graphs. Algorithmica 20(3) (1998), 277–293.
P. M. Gruber and J. M. Wills: Handbook of Convex Geometry, North-Holland, Amsterdam, 1993.
J. Kingman: Poisson Processes, Oxford University Press, Oxford, 1993.
T. Łuczak: The chromatic number of random graphs, Combinatorica 11(1) (1991), 45–54.
C. L. Mallows: An inequality involving multinomial probabilities. Biometrika 55(2) (1968), 422–424.
J. Matoušek: Lectures on Discrete Geometry, volume 212 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2002.
C. J. H. McDiarmid: Random channel assignment in the plane, Random Structures and Algorithms 22(2) (2003), 187–212.
T. Müller: Two-point concentration in random geometric graphs. Combinatorica 28(5) (2008), 529–545.
J. Pach and P. K. Agarwal: Combinatorial Geometry, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons Inc., New York, 1995. A Wiley-Interscience Publication.
R. Peeters: On coloring j-unit sphere graphs. Technical Report FEW 512, Economics Department, Tilburg University, 1991.
M. D. Penrose: Random Geometric Graphs, Oxford University Press, Oxford, 2003.
V. Raghavan and J. Spinrad: Robust algorithms for restricted domains, J. Algorithms 48(1) (2003), 160–172. Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms (Washington, DC, 2001).
C. A. Rogers: Packing and Covering, Cambridge Tracts in Mathematics and Mathematical Physics, No. 54. Cambridge University Press, New York, 1964.
E. R. Scheinerman and D. H. Ullman: Fractional Graph Theory, Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons Inc., New York, 1997.
J. H. van Lint and R. M. Wilson: A Course in Combinatorics, Cambridge University Press, Cambridge, second edition, 2001.
Author information
Authors and Affiliations
Corresponding author
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.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-011-2403-3