Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-02T16:48:43.206Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Achlioptas, D. and Moore, C. (2004) The chromatic number of random regular graphs. In Proc. RANDOM'04, pp. 219–228.Google Scholar
[2]Achlioptas, D. and Naor, A. (2005) The two possible values of the chromatic number of a random graph. Ann. of Math. 162 1335–1351.CrossRefGoogle Scholar
[3]Billingsley, P. (1986) Probability and Measure, 2ndedn, Wiley, New York.Google Scholar
[4]Bollobás, B. (1985) Random Graphs, Academic Press New York.Google Scholar
[5]Borodin, O. and Kostochka, A. (1977) On an upper bound of a graph's chromatic number depending on the graph's degree and density. J. Combin. Theory Ser. B 23 247–250.CrossRefGoogle Scholar
[6]Bressan, A. (1988) Unique solutions for a class of discontinuous differential equations. Proc. Amer. Math. Soc. 104 772–778.Google Scholar
[7]Catlin, P. (1978) A bound on the chromatic number of a graph. Discrete Math. 22 81–83.Google Scholar
[8]Cooper, C., Frieze, A., Reed, B. and Riordan, O. (2002) Random regular graphs of non-constant degree: Independence and chromatic number. Combin. Probab. Comput. 11 323–342.CrossRefGoogle Scholar
[9]Díaz, J., Serna, M. and Wormald, N. C. Computation of the bisection width for random d-regular graphs. Theoret. Comput. Sci., to appear.Google Scholar
[10]Díaz, J., Grammatikopoulos, G., Kaporis, A. C., Kirousis, L. M., Perez, X. and Sotiropoulos, D. G. (2005) 5-regular multi-graphs are 3-colorable with positive probability independent of their size. To appear in Proc. 13th Annual European Symposium on Algorithms.Google Scholar
[11]Frieze, A. M. and Luczak, T. (1992) On the independence and chromatic numbers of random regular graphs. J. Combin. Theory Ser. B 54 123–132.CrossRefGoogle Scholar
[12]Kemkes, G. and Wormald, N. (2005) On the chromatic number of a random 5-regular graph. Department of C&O research report series CORR 2005-22, University of Waterloo.Google Scholar
[13]Kostochka, A. V. (1978) Degree, girth and chromatic number. In Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976, Vol. II, Colloq. Math. Soc. Janos Bolyai, Vol.18, North-Holland, Amsterdam/New York. pp. 679–696.Google Scholar
[14]Krivelevich, M., Sudakov, B., Vu, V. H. and Wormald, N. C. (2001) Random regular graphs of high degree. Random Struct. Alg. 18 346–363.CrossRefGoogle Scholar
[15]Lawrence, J. (1978) Covering the vertex set of a graph with subgraphs of smaller degree. Discrete Math. 21 61–68.CrossRefGoogle Scholar
[16]Molloy, M. (1992) Mathematics thesis, University of Waterloo.Google Scholar
[17]Shi, L. and Wormald, N. (2007) Colouring random 4-regular graphs. Combin. Probab. Comput. 16 309–344.Google Scholar
[18]Wormald, N. C. (1995) Differential equations for random processes and random graphs. Ann. Appl. Probab. 5 1217–1235.CrossRefGoogle Scholar
[19]Wormald, N. C. (1999) The differential equation method for random graph processes and greedy algorithms. In Lectures on Approximation and Randomized Algorithms (Karoński, M. and Prömel, H., eds), PWN, Warsaw, pp. 73–155.Google Scholar
[20]Wormald, N. C.(1999) Models of random regular graphs. In Surveys in Combinatorics, 1999 (Lamb, J. D. and Preece, D. A.eds), pp. 239–298.Google Scholar
[21]Wormald, N. C. (2003) Analysis of greedy algorithms on graphs with bounded degrees. Discrete Math. 273 235–260.CrossRefGoogle Scholar