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Colouring Random Regular Graphs

Published online by Cambridge University Press:  01 May 2007

LINGSHENG SHI  and
NICHOLAS WORMALD
LINGSHENG SHI
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo ON, CanadaN2L 3G1 (e-mail: lshi@uwaterloo.ca, nwormald@uwaterloo.ca)
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Abstract

In a previous paper we showed that a random 4-regular graph asymptotically almost surely (a.a.s.) has chromatic number 3. Here we extend the method to show that a random 6-regular graph asymptotically almost surely (a.a.s.) has chromatic number 4 and that the chromatic number of a random d-regular graph for other d between 5 and 10 inclusive is a.a.s. restricted to a range of two integer values: {3, 4} for d = 5, {4, 5} for d = 7, 8, 9, and {5, 6} for d = 10. The proof uses efficient algorithms which a.a.s. colour these random graphs using the number of colours specified by the upper bound. These algorithms are analysed using the differential equation method, including an analysis of certain systems of differential equations with discontinuous right-hand sides.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 16 , Issue 3 , May 2007 , pp. 459 - 494
Copyright
Copyright © Cambridge University Press 2006

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