Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, March 16, 2006, 12:15 pm

Duration: This information is not available in the database

Location: This information is not available in the database

Speaker: Tobias Müller (Univ. of Oxford)

Two-point concentration in random geometric graphs

A random geometric graph G_n is constructed by taking vertices X₁, ..., X_n ∈ R^d at random (i.i.d. according to some probability distribution ν) 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 stating that when n r^d = o(ln n) then the probability distribution of the clique number ω(G_n) becomes concentrated on two consecutive integers in the sense that P(ω(G_n) ∈ {k(n), k(n)+1} ) tends to 1 for some sequence k(n).

We also show that the same holds for a number of other graph parameters including the chromatic number Χ(G_n). A series of celebrated results establish that a similar phenomenon occurs in the Erdős-Rényi or G(n,p)-model of random graphs.

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