Colouring random regular graphs

Lingsheng Shi, Nicholas Wormald

Research output: Contribution to journalArticleResearchpeer-review

7 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)459-494
Number of pages36
JournalCombinatorics, Probability and Computing
Issue number3
Publication statusPublished - May 2007
Externally publishedYes

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