Advertisement

On the Chromatic Number of Random Graphs

  • Amin Coja-Oghlan
  • Konstantinos Panagiotou
  • Angelika Steger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4596)

Abstract

In this paper we study the chromatic number χ(G n,p) of the binomial random graph G n,p, where p = p(n) ≤ n  − 3/4 − δ , for every fixed δ> 0. We prove that a.a.s. χ(G n,p) is ℓ, ℓ + 1, or ℓ + 2, where ℓ is the maximum integer satisfying 2(ℓ − 1)log(ℓ − 1) ≤ np.

Keywords

Random Graph Average Degree Chromatic Number Color Classis Additional Color 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Feige, U., Kilian, J.: Zero knowledge and the chromatic number. J. Comput. System Sci. 57(2), 187–199 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Zeitlhofer, T., Wess, B.: A comparison of graph coloring heuristics for register allocation based on coalescing in interval graphs. In: ISCAS (4), 529–532 (2004)Google Scholar
  3. 3.
    Johnson, D.S., Aragon, C.R., McGeoch, L.A., Schevon, C.: Optimization by simulated annealing: an experimental evaluation; part ii, graph coloring and number partitioning. Oper. Res. 39(3), 378–406 (1991)zbMATHCrossRefGoogle Scholar
  4. 4.
    Krivelevich, M.: Coloring random graphs—an algorithmic perspective. In: Mathematics and computer science, II (Versailles, 2002). Trends Math. Birkhäuser, Basel, pp. 175–195 (2002)Google Scholar
  5. 5.
    Achlioptas, D., Moore, C.: Almost all graphs with average degree 4 are 3-colorable. J. Comput. System Sci. 67(2), 441–471 (2003), Special issue on STOC 2002 (Montreal, QC)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Jia, H., Moore, C.: How much backtracking does it take to color random graphs? Rigorous results on heavy tails (2004)Google Scholar
  7. 7.
    Mulet, R., Pagnani, A., Weigt, M., Zecchina, R.: Coloring random graphs. Physical Review Letters 89, 268701 (2002)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Braunstein, A., Mulet, R., Pagnani, A., Weigt, M., Zecchina, R.: Polynomial iterative algorithms for coloring and analyzing random graphs. Physical Review E 68, 36702 (2003)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Achlioptas, D., Naor, A.: The two possible values of the chromatic number of a random graph. Ann. of Math. (2) 162(3), 1335–1351 (2005)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Erdős, P., Rényi, A.: On random graphs. I. Publ. Math. Debrecen 6, 290–297 (1959)Google Scholar
  11. 11.
    Bollobás, B.: The chromatic number of random graphs. Combinatorica 8(1), 49–55 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Łuczak, T.: A note on the sharp concentration of the chromatic number of random graphs. Combinatorica 11(3), 295–297 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Shamir, E., Spencer, J.: Sharp concentration of the chromatic number on random graphs G n,p. Combinatorica 7(1), 121–129 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Alon, N., Krivelevich, M.: The concentration of the chromatic number of random graphs. Combinatorica 17(3), 303–313 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Achlioptas, D., Friedgut, E.: A sharp threshold for k-colorability. Random Structures Algorithms 14(1), 63–70 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
  17. 17.
    Alon, N., Spencer, J.H.: The probabilistic method, 2nd edn. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience [John Wiley & Sons], Chichester (2000)zbMATHGoogle Scholar
  18. 18.
    Janson, S., Łuczak, T., Rucinski, A.: Random graphs. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, Chichester (2000)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Amin Coja-Oghlan
    • 1
  • Konstantinos Panagiotou
    • 2
  • Angelika Steger
    • 2
  1. 1.Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA15213USA
  2. 2.Institute of Theoretical Computer Science ETH Zentrum, Universitätsstr. 6, CH - 8092 ZurichSwitzerland

Personalised recommendations