Two-point concentration in random geometric graphs

A random geometric graph Gn is constructed by taking vertices X1, . . . , Xn Rd at random (i.i.d. according to some probability distribution with a bounded density function) and including an edge between Xi and Xj if Xi -Xj < r where r = r(n) > 0. We prove a conjecture of Penrose ([14]) stating that when r = r(n) is chosen such that nrd = o(ln n) then the probability distribution of the clique number (Gn) becomes concentrated on two consecutive integers and we show that the same holds for a number of other graph parameters including the chromatic number (Gn).

• Type: Journal
• Year: 2008
• Where: COMBINATORICA
• Authors: Tobias Müller
• Keywords: COMBINATORICA 2008, Geometric Graph Gn, Probability Distribution, Xn Rd