Two-point concentration in random geometric graphs

Abstract

A random geometric graph G n is constructed by taking vertices X 1,…,X n ∈ℝd at random (i.i.d. according to some probability distribution ν with a bounded density function) and including an edge between X i and X j if ‖X i -X j ‖ < r where r = r(n) > 0. We prove a conjecture of Penrose ([14]) stating that when r=r(n) is chosen such that nr d = o(lnn) then the probability distribution of the clique number ω(G n ) becomes concentrated on two consecutive integers and we show that the same holds for a number of other graph parameters including the chromatic number χ(G n ).

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Correspondence to Tobias Müller.

Additional information

The author was partially supported by EPSRC, the Department of Statistics, Bekkerla-Bastide fonds, Dr. Hendrik Muller’s Vaderlandsch fonds, and Prins Bernhard Cultuurfonds.

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Müller, T. Two-point concentration in random geometric graphs. Combinatorica 28, 529 (2008). https://doi.org/10.1007/s00493-008-2283-3

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Mathematics Subject Classification (2000)

  • 05C80
  • 05C15
  • 05C69
  • 60D05