## DMTCS Proceedings, 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)

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DMTCS Conference vol AE (2005), pp. 1-4

## 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)

### DMTCS Conference Volume AE (2005), pp. 1-4

author: Colin J. H. McDiarmid and Tobias Müller Colouring random geometric graphs random geometric graphs, graph colouring A random geometric graph G n is obtained as follows. We take X 1 , X 2 , …, X n ∈ℝ d at random (i.i.d. according to some probability distribution ν on ℝ d ). For i ≠j we join X i and X j by an edge if ║X i - X j ║< r(n) . We study the properties of the chromatic number χ n and clique number ω n 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 d . 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 n r d (which is a measure of the average degree). If r(n) is such that n r d  /  ln n →0 as n →∞ then χ n  / ω n →1 almost surely. If n r d  /  ln n →∞ then χ n  / ω n →1 / δ almost surely, where δ is the (translational) packing density of the unit ball B := { x ∈ℝ d : ║x║< 1 } (i.e. δ is the proportion of d -space that can be filled with disjoint translates of B ). If n r d  /  ln n →t ∈(0,∞) then χ n  / ω n 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 χ n and  ω n . We are also able to prove a conjecture by Penrose. This states that when n r d  /  ln n →0 then the clique number becomes focussed on two adjacent integers, meaning that there exists a sequence k(n) such that P ( ω n ∈{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). If your browser does not display the abstract correctly (because of the different mathematical symbols) you may look it up in the PostScript or PDF files. Colin J. H. McDiarmid and Tobias Müller (2005), Colouring random geometric graphs, in 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AE, pp. 1-4 For a corresponding BibTeX entry, please consider our BibTeX-file. dmAE0101.ps.gz (71 K) dmAE0101.ps (164 K) dmAE0101.pdf (154 K)