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Abstract

If each edge (u,v) of a graph G = (V,E) is decorated with a permutation π u,v of k objects, we say that it has a permuted k-coloring if there is a coloring σ:V → {1,…,k} such that σ(v) ≠ π u,v (σ(u)) for all (u,v) ∈ E. Based on arguments from statistical physics, we conjecture that the threshold d k for permuted k-colorability in random graphs G(n,m = dn/2), where the permutations on the edges are uniformly random, is equal to the threshold for standard graph k-colorability. The additional symmetry provided by random permutations makes it easier to prove bounds on d k . By applying the second moment method with these additional symmetries, and applying the first moment method to a random variable that depends on the number of available colors at each vertex, we bound the threshold within an additive constant. Specifically, we show that for any constant ε > 0, for sufficiently large k we have

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In contrast, the best known bounds on d k for standard k-colorability leave an additive gap of about ln k between the upper and lower bounds.

Keywords

Random Graph Chromatic Number Random Permutation Isoperimetric Inequality Moment Method 
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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References

  1. 1.
    Achlioptas, D., Coja-Oghlan, A., Ricci-Tersenghi, F.: On the solution-space geometry of random constraint satisfaction problems. Random Struct. Algorithms 38(3), 251–268 (2011)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Achlioptas, D., Molloy, M.: Almost All Graphs with 2.522 n Edges are not 3-Colorable. Electronic Journal of Combinatorics 6 (1999)Google Scholar
  3. 3.
    Achlioptas, D., Moore, C.: Two moments suffice to cross a sharp threshold. SIAM Journal on Computing 36, 740–762 (2006)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Achlioptas, D., Moore, C.: On the 2-Colorability of Random Hypergraphs. In: Rolim, J.D.P., Vadhan, S.P. (eds.) RANDOM 2002. LNCS, vol. 2483, pp. 78–90. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Achlioptas, D., Moore, C.: The Chromatic Number of Random Regular Graphs. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) APPROX and RANDOM 2004. LNCS, vol. 3122, pp. 219–228. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  6. 6.
    Achlioptas, D., Naor, A.: The Two Possible Values of the Chromatic Number of a Random Graph. Ann. Math. 162(3), 1333–1349 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Achlioptas, D., Peres, Y.: The Threshold for Random k-SAT is 2k log2 − O(k). J. AMS 17, 947–973 (2004)MathSciNetMATHGoogle Scholar
  8. 8.
    Bhatnagar, N., Vera, J.C., Vigoda, E., Weitz, D.: Reconstruction for Colorings on Trees. SIAM J. Discrete Math. 25(2), 809–826 (2011)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Coja-Oghlan, A., Panagiotou, K.: Catching the k-NAESAT threshold. In: Proc. STOC 2012, pp. 899–908 (2012)Google Scholar
  10. 10.
    Coja-Oghlan, A., Zdeborová, L.: The condensation transition in random hypergraph 2-coloring. In: Proc. SODA 2012, pp. 241–250 (2012)Google Scholar
  11. 11.
    Dubois, O., Mandler, J.: On the non-3-colorability of random graphs (preprint), arXiv:math/0209087v1Google Scholar
  12. 12.
    Kaporis, A.C., Kirousis, L.M., Stamatiou, Y.C.: A note on the non-colorability threshold of a random graph. Electronic Journal of Combinatorics 7(1) (2000)Google Scholar
  13. 13.
    Krząkala, F., Montanari, A., Ricci-Tersenghi, F., Semerjian, G., Zdeborová, L.: Gibbs states and the set of solutions of random constraint satisfaction problems. Proc. Natl. Acad. Sci. 104(25), 10318–10323 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Krząkala, F., Zdeborová, L.: Potts Glass on Random Graphs. Euro. Phys. Lett. 81, 57005 (2008)CrossRefGoogle Scholar
  15. 15.
    Maneva, E.N., Sinclair, A.: On the satisfiability threshold and clustering of solutions of random 3-SAT formulas. Theor. Comp. Sci. 407(1-3), 359–369 (2008)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Mertens, S., Mézard, M., Zecchina, R.: Threshold values of Random k-SAT from the cavity method. Random Structures and Algorithms 28, 340–373 (2006)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Mézard, M., Parisi, G., Zecchina, R.: Analytic and Algorithmic Solution of Random Satisfiability Problems. Science 297 (2002)Google Scholar
  18. 18.
    Mitzenmacher, M., Upfal, E.: Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press (2005)Google Scholar
  19. 19.
    Montanari, A., Restrepo, R., Tetali, P.: Reconstruction and Clustering in Random Constraint Satisfaction Problems. SIAM J. Disc. Math. 25(2), 771–808 (2011)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Mulet, R., Pagnani, A., Weigt, M., Zecchina, R.: Coloring random graphs. Phys. Rev. Lett. 89 (2002)Google Scholar
  21. 21.
    Sly, A.: Reconstruction of Random Colourings. Communications in Mathematical Physics 288(3), 943–961 (2009)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Zdeborová, L., Boettcher, S.: Conjecture on the maximum cut and bisection width in random regular graphs. J. Stat. Mech. (2010)Google Scholar
  23. 23.
    Zdeborová, L., Krząkala, F.: Phase transitions in the coloring of random graphs. Phys. Rev. E 76, 031131 (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Varsha Dani
    • 1
  • Cristopher Moore
    • 1
    • 2
  • Anna Olson
    • 3
  1. 1.Computer Science DepartmentUniversity of New MexicoUSA
  2. 2.Santa Fe InstituteUSA
  3. 3.Computer Science DepartmentUniversity of ChicagoUSA

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