Colouring random geometric graphs

A random geometric graph Gn is obtained as follows. We takeX1, X2, . . . , Xn ∈ R at random (i.i.d. according to some probability distribution ν on R). For i 6= j we joinXi andXj by an edge if ‖Xi−Xj ‖< r(n). We study the properties of the chromatic number χn and clique number ωn of this graph as n becomes large, where we assume that r(n) → 0. We allow any choice ν that has a bounded density function and ‖ .‖ may be any norm on R. Depending on the choice of r(n), qualitatively different types of behaviour can be observed. We distinguish three main cases, in terms of the key quantity nr (which is a measure of the average degree). If r(n) is such that nr d lnn → 0 as n → ∞ then χn ωn → 1 almost surely. If nr d lnn →∞ then χn ωn → 1 δ almost surely, where δ is the (translational) packing density of the unit ball B := {x ∈ R :‖x‖< 1} (i.e. δ is the proportion of d-space that can be filled with disjoint translates ofB). If nr d lnn → t ∈ (0,∞) then χn ωn tends almost surely to a constant that can be bounded in terms of δ and t. These results extend earlier work of McDiarmid and Penrose. The proofs in fact yield separate expressions for χn and ωn. We are also able to prove a conjecture by Penrose. This states that when nr d lnn → 0 then the clique number becomes focussed on two adjacent integers, meaning that there exists a sequence k(n) such that P(ωn ∈ {k(n), k(n)+1}) → 1 as n → ∞. The analogous result holds for the chromatic number (and for the maximum degree, as was already shown by Penrose in the uniform case).