Random Regular Graphs of Non-Constant Degree: Independence and Chromatic Number

Authors:
Colin Cooper;Alan Frieze;Bruce Reed;Oliver Riordan
Affiliations:
Department of Mathematical and Computing Sciences, Goldsmiths College, University of London, New Cross, London SE14 6NW, England (e-mail: c.cooper@gold.ac.uk);Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: alan@random.math.cmu.edu);Equipe Combinatoire, CNRS, Université de Paris VI, 4 Place Jussieu, Paris 75005, France (e-mail: reed@ecp6.jussieu.fr);Trinity College, Cambridge CB2 1TQ, England (e-mail: omr10@dpmms.cam.ac.uk)
Venue:
Combinatorics, Probability and Computing
Year:
2002

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Abstract

Let r = r(n) → ∞ with 3 ⩽ r ⩽ n1−η for an arbitrarily small constant η 0, and let Gr denote a graph chosen uniformly at random from the set of r-regular graphs with vertex set {1, 2, …, n}. We prove that, with probability tending to 1 as n → ∞, Gr has the following properties: the independence number of Gr is asymptotically 2n log r/r and the chromatic number of Gr is asymptotically r/2nlogr.