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On the Chromatic Number of Random Graphs with a Fixed Degree Sequence

Published online by Cambridge University Press:  01 September 2007

ALAN FRIEZE ,
MICHAEL KRIVELEVICH  and
CLIFF SMYTH
ALAN FRIEZE
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA15213, USA (e-mail: [email protected])
MICHAEL KRIVELEVICH
Affiliation:
Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: [email protected])
CLIFF SMYTH
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (e-mail: [email protected])
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Abstract

Let d=1≤d1≤ d2≤···.≤ dn be a non-decreasing sequence of n positive integers, whose sum is even. Let denote the set of graphs with vertex set [n]={1,2,. . .., n} in which the degree of vertex i is di. Let Gn,d be chosen uniformly at random from . Let d=(d1+d2+···.+dn)/n be the average degree. We give a condition on d under which we can show that w.h.p. the chromatic number of is Θ(d/ln d). This condition is satisfied by graphs with exponential tails as well those with power law tails.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 16 , Issue 5 , September 2007 , pp. 733 - 746
Copyright
Copyright © Cambridge University Press 2007

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References

[1]Achlioptas, D. and Moore, C. (2004) The chromatic number of random regular graphs. In Proc. RANDOM 2004, 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. 62 1335–1351.CrossRefGoogle Scholar
[3]Aiello, W., Chung, F. and Lu, L. (2001) A random graph model for power law graphs. Experiment. Math. 10 53–66.CrossRefGoogle Scholar
[4]Alon, N., Krivelevich, M. and Sudakov, B. (1999) Coloring graphs with sparse neighborhoods. J. Combin. Theory Ser. B 77 73–82.CrossRefGoogle Scholar
[5]Bender, E. A. and Canfield, E. R. (1978) The asymptotic number of labelled graphs with given degree sequences. J. Combin. Theory Ser. A 24 296–307.CrossRefGoogle Scholar
[6]Bollobás, B. (1980) A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. Europ. J. Combin. 1 311–316.CrossRefGoogle Scholar
[7]Bollob´s, B. (1988) The chromatic number of random graphs. Combinatorica 8 49–55.CrossRefGoogle Scholar
[8]Cooper, C. and Frieze, A. M. (2004) The size of the largest strongly connected component of a random digraph with a given degree sequence. Combin. Probab. Comput. 13 319–338.CrossRefGoogle Scholar
[9]Cooper, C., Frieze, A. M., Reed, B. A. and Riordan, O. (2002) Random regular graphs of non-constant degree: Independence and chromatic number. Combin. Probab. Comput. 11 323–342.CrossRefGoogle Scholar
[10]Frieze, A. M. and Łuczak, T. (1992) On the independence and chromatic numbers of random regular graphs. J. Combin. Theory Ser. B 54 123–132.Google Scholar
[11]Frieze, A. M. and Pittel, B. G. (2004) Perfect matchings in random graphs with prescribed minimal degree. In Mathematics and Computer Science III (Drmota, M., Flajolet, P., Gardy, D. and Gittenberger, B., eds), Trends in Mathematics, Birkhäuser, Basel, pp. 95–132.CrossRefGoogle Scholar
[12]Łuczak, T. (1991) The chromatic number of random graphs. Combinatorica 11 45–54.CrossRefGoogle Scholar
[13]McKay, B. D. and Wormald, N. C. (1990) Asymptotic enumeration by degree sequence of graphs of high degree. Europ. J. Combin. 11 565–580.CrossRefGoogle Scholar
[14]McKay, B. D. and Wormald, N. C. (1990) Uniform generation of random regular graphs of moderate degree. J. Algorithms 11 52–67.CrossRefGoogle Scholar
[15]McKay, B. D. and Wormald, N. C. (1991) Asymptotic enumeration by degree sequence of graphs with degree o(n 1/2). Combinatorica 11 369–382.CrossRefGoogle Scholar
[16]Molloy, M. and Reed, B. A. (1995) A critical point for random graphs with a given degree sequence. Random Struct. Alg. 6 161–180.CrossRefGoogle Scholar
[17]Molloy, M. and Reed, B. A. (1998) The size of the largest component of a random graph on a fixed degree sequence. Combin. Probab. Comput. 7 295–306.Google Scholar
[18]Wormald, N. C. (1999) Models of random regular graphs. In Surveys in Combinatorics: Proc. 1999 British Combinatorial Conference (Lamb, J. D. and Preece, D. A., eds), Vol. 267 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, pp. 239–298.CrossRefGoogle Scholar