On the independence and chromatic numbers of random regular graphs
Journal of Combinatorial Theory Series B
Random regular graphs of high degree
Random Structures & Algorithms
Random Regular Graphs of Non-Constant Degree: Independence and Chromatic Number
Combinatorics, Probability and Computing
Colouring Random 4-Regular Graphs
Combinatorics, Probability and Computing
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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.