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