Why Almost All k-Colorable Graphs Are Easy

  • Amin Coja-Oghlan
  • Michael Krivelevich
  • Dan Vilenchik
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4393)


Coloring a k-colorable graph using k colors (k ≥ 3) is a notoriously hard problem. Considering average case analysis allows for better results. In this work we consider the uniform distribution over k-colorable graphs with n vertices and exactly cn edges, c greater than some sufficiently large constant. We rigorously show that all proper k-colorings of most such graphs are clustered in one cluster, and agree on all but a small, though constant, number of vertices. We also describe a polynomial time algorithm that finds a proper k-coloring for (1 − o(1))-fraction of such random k-colorable graphs, thus asserting that most of them are “easy”. This should be contrasted with the setting of very sparse random graphs (which are k-colorable whp), where experimental results show some regime of edge density to be difficult for many coloring heuristics. One explanation for this phenomena, backed up by partially non-rigorous analytical tools from statistical physics, is the complicated clustering of the solution space at that regime, unlike the more “regular” structure that denser graphs possess. Thus in some sense, our result rigorously supports this explanation.


Exchange Rate Random Graph Polynomial Time Algorithm Chromatic Number Graph Property 
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.


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  1. 1.
    Achlioptas, D., Friedgut, E.: A sharp threshold for k-colorability. Random Struct. Algorithms 14(1), 63–70 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Achlioptas, D., Molloy, M.: Almost all graphs with 2.522n edges are not 3-colorable. Elec. Jour. Of Comb. 6(1), R29 (1999)MathSciNetGoogle Scholar
  3. 3.
    Achlioptas, D., Moore, C.: Almost all graphs with average degree 4 are 3-colorable. In: STOC ’02, pp. 199–208 (2002)Google Scholar
  4. 4.
    Alon, N., Kahale, N.: A spectral technique for coloring random 3-colorable graphs. SIAM J. on Comput. 26(6), 1733–1748 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Ben-Sasson, E., Bilu, Y., Gutfreund, D.: Finding a randomly planted assignment in a random 3CNF. Manuscript (2002)Google Scholar
  6. 6.
    Blum, A., Spencer, J.: Coloring random and semi-random k-colorable graphs. J. of Algorithms 19(2), 204–234 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Bollobás, B.: The chromatic number of random graphs. Combin. 8(1), 49–55 (1988)CrossRefzbMATHGoogle Scholar
  8. 8.
    Böttcher, J.: Coloring sparse random k-colorable graphs in polynomial expected time. In: Proc. 30th MFCS, pp. 156–167 (2005)Google Scholar
  9. 9.
    Braunstein, A., et al.: Constraint satisfaction by survey propagation. In: Computational Complexity and Statistical Physics (2005)Google Scholar
  10. 10.
    Chen, H.: An algorithm for sat above the threshold. In: 6th International Conference on Theory and Applications of Satisfiability Testing, pp. 14–24 (2003)Google Scholar
  11. 11.
    Coja-Oghlan, A.: Coloring semirandom graphs optimally. In: Díaz, J., et al. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 383–395. Springer, Heidelberg (2004)Google Scholar
  12. 12.
    Dyer, M.E., Frieze, A.M.: The solution of some random NP-hard problems in polynomial expected time. J. Algorithms 10(4), 451–489 (1989)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Feige, U., Kilian, J.: Zero knowledge and the chromatic number. J. Comput. and Syst. Sci. 57(2), 187–199 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Feige, U., Kilian, J.: Heuristics for semirandom graph problems. J. Comput. and Syst. Sci. 63(4), 639–671 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Feige, U., Mossel, E., Vilenchik, D.: Complete convergence of message passing algorithms for some satisfiability problems. In: RANDOM, pp. 339–350 (2006)Google Scholar
  16. 16.
    Flaxman, A.: A spectral technique for random satisfiable 3CNF formulas. In: Proc. 14th ACM-SIAM Symp. on Discrete Algorithms, pp. 357–363. ACM Press, New York (2003)Google Scholar
  17. 17.
    Frieze, A., Jerrum, M.: Improved approximation algorithms for MAX k-CUT and MAX BISECTION. Algorithmica 18(1), 67–81 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric algorithms and combinatorial optimization, 2nd edn. Algorithms and Combinatorics, vol. 2. Springer, Berlin (1993)zbMATHGoogle Scholar
  19. 19.
    Krivelevich, M., Vilenchik, D.: Semirandom models as benchmarks for coloring algorithms. In: ANALCO, pp. 211–221 (2006)Google Scholar
  20. 20.
    Kučera, L.: Expected behavior of graph coloring algorithms. In: Karpinski, M. (ed.) FCT 1977. LNCS, vol. 56, pp. 447–451. Springer, Heidelberg (1977)Google Scholar
  21. 21.
    Łuczak, T.: The chromatic number of random graphs. Combin. 11(1), 45–54 (1991)CrossRefzbMATHGoogle Scholar
  22. 22.
    Mulet, R., et al.: Coloring random graphs. Phys. Rev. Lett. 89(26), 268701 (2002)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Prömel, H., Steger, A.: Random l-colorable graphs. Random Structures and Algorithms 6, 21–37 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Turner, J.S.: Almost all k-colorable graphs are easy to color. J. Algorithms 9(1), 63–82 (1988)CrossRefMathSciNetzbMATHGoogle Scholar

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© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Amin Coja-Oghlan
    • 1
  • Michael Krivelevich
    • 2
  • Dan Vilenchik
    • 3
  1. 1.Institute for Informatics, Humboldt-University, BerlinGermany
  2. 2.School Of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-AvivIsrael
  3. 3.School of Computer Science, Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-AvivIsrael

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