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Random Geometric Graphs$
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Mathew Penrose

Print publication date: 2003

Print ISBN-13: 9780198506263

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198506263.001.0001

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MAXIMUM DEGREE, CLIQUES, AND COLOURINGS

MAXIMUM DEGREE, CLIQUES, AND COLOURINGS

Chapter:
(p.109) 6 MAXIMUM DEGREE, CLIQUES, AND COLOURINGS
Source:
Random Geometric Graphs
Author(s):

Mathew Penrose (Contributor Webpage)

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198506263.003.0006

This chapter is concerned with maximum degree, clique number, and chromatic number. A focusing (i.e., two-point concentration) phenomenon is demonstrated for the distribution of the maximum degree in the subconnective regime, and likewise for clique number. Laws of large numbers are presented for maximum degree and clique number in the subconnective and connectivity regimes. For the chromatic number, a law of large numbers is presented in the subconnectivity regime, and bounds on its ratio to the clique number are given for the superconnective regime. These bounds are given in terms of packings of the unit ball.

Keywords:   maximum degree, clique number, chromatic number, focusing, large numbers, packing

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