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Pré-Publication, Document De Travail Année : 2013

Clique number of random geometric graphs

Résumé

The clique number C of a graph is the largest clique size in the graph. For a random geometric graph of n vertices, taken uniformly at random, including an edge beween two vertices if their distance, taken with the uniform norm, is less than a parameter r on a torus Tda, we find the asymptotic behaviour of the clique number. Setting θ = (r)d, in the a subcritical regime where θ = o( 1 ), we exhibit the intervals of θ where C n takes the same value asymptotically almost surely. In the critical regime, θ ∼ 1 , we show that C is growing slightly slower than ln n asymptotically n almost surely. Finally, in the supercritical regime, 1 = o(θ), we prove n that C grows as nθ asymptotically almost surely. We also investigate the behaviour of related graph characteristics: the chromatic number, the maximum vertex degree, and the independence number.
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Dates et versions

hal-00864303 , version 1 (23-09-2013)
hal-00864303 , version 2 (05-12-2013)
hal-00864303 , version 3 (21-03-2014)
hal-00864303 , version 4 (04-09-2017)

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  • HAL Id : hal-00864303 , version 1

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Laurent Decreusefond, Philippe Martins, Anaïs Vergne. Clique number of random geometric graphs. 2013. ⟨hal-00864303v1⟩
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