Local convergence of random graph colorings

  • Amin Coja-Oghlan
  • Charilaos Efthymiou
  • Nor Jaafari

Mathematics Subject Classification (2000)

05C80 05C15 


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  1. [1]
    D. Achlioptas and E. Friedgut: A sharp threshold for k-colorability, Random Struct. Algorithms 14 (1999), 63–70.MathSciNetMATHGoogle Scholar
  2. [2]
    D. Achlioptas and A. Coja-Oghlan: Algorithmic barriers from phase transitions, Proc. 49th FOCS (2008), 793–802.Google Scholar
  3. [3]
    D. Achlioptas and M. Molloy: The analysis of a list-coloring algorithm on a random graph, Proc. 38th FOCS (1997), 204–212.Google Scholar
  4. [4]
    D. Achlioptas and A. Naor: The two possible values of the chromatic number of a random graph, Annals of Mathematics 162 (2005), 1333–1349.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    D. Aldous and J. Steele: The objective method: probabilistic combinatorial op- timization and local weak convergence (2003), in: Probability on discrete structures (H. Kesten (ed.)), Springer 2004.Google Scholar
  6. [6]
    N. Alon and M. Krivelevich: The concentration of the chromatic number of random graphs, Combinatorica 17 (1997), 303–313.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    V. Bapst, A. Coja-Oghlan and C. Efthymiou: Planting colourings silently, arXiv:1411.0610 (2014).Google Scholar
  8. [8]
    V. Bapst, A. Coja-Oghlan, S. Hetterich, F. Rassmann and Dan Vilenchik: The condensation phase transition in random graph coloring, Communications in Mathematical Physics 341 (2016), 543–606.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    I. Benjamini and O. Schramm: Recurrence of distributional limits of nite planar graphs, Electronic J. Probab. 6 (2001), 1–13.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    N. Bhatnagar, J. Vera, E. Vigoda and D. Weitz: Reconstruction for Colorings on Trees, SIAM Journal on Discrete Mathematics. 25 (2011), 809–826.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    N. Bhatnagar, A. Sly and P. Tetali: Decay of correlations for the hardcore model on the d-regular random graph, Electronic Journal of Probability 21 (2016).Google Scholar
  12. [12]
    B. Bollobás: The chromatic number of random graphs, Combinatorica 8 (1988), 49–55.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    C. Bordenave and P. Caputo: Large deviations of empirical neighborhood distri- bution in sparse random graphs, Probability Theory and Related Fields 163 (2015), 149–222.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    A. Coja-Oghlan: Upper-bounding the k-colorability threshold by counting covers, Electronic Journal of Combinatorics 20 (2013), P32.MathSciNetMATHGoogle Scholar
  15. [15]
    A. Coja-Oghlan and D. Vilenchik: The chromatic number of random graphs for most average degrees, International Mathematics Research Notices, (2015): rnv333.Google Scholar
  16. [16]
    M. E. Dyer and A. M. Frieze: Randomly coloring random graphs, Random Struct. Algorithms 36 (2010), 251–272.MathSciNetMATHGoogle Scholar
  17. [17]
    M. Dyer, A. Flaxman, A. Frieze and E. Vigoda: Randomly coloring sparse random graphs with fewer colors than the maximum degree, Random Structures and Algorithms 29 (2006), 450–465.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    C. Efthymiou: MCMC sampling colourings and independent sets of G(n;d=n) near uniqueness threshold, Proc. 25th SODA (2014), 305–316.Google Scholar
  19. [19]
    C. Efthymiou: Switching colouring of G(n;d=n) for sampling up to Gibbs unique- ness threshold, Proc. 22nd ESA (2014), 371–381.Google Scholar
  20. [20]
    C. Efthymiou: Reconstruction/non-reconstruction thresholds for colourings of gen- eral Galton-Watson trees, Proc. 19th RANDOM (2015), 756–774.Google Scholar
  21. [21]
    P. Erdős: Graph theory and probability, Canad. J. Math. 11 (1959), 34–38.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    P. Erdős and A. Rényi: On the evolution of random graphs, Magayar Tud. Akad. Mat. Kutato Int. Kozl. 5 (1960), 17–61.MathSciNetMATHGoogle Scholar
  23. [23]
    A. Gerschenfeld and A. Montanari: Reconstruction for models on random graphs, Proc. 48th FOCS (2007), 194–204.Google Scholar
  24. [24]
    G. Grimmett and C. McDiarmid: On colouring random graphs, Mathematical Proceedings of the Cambridge Philosophical Society 77 (1975), 313–324.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    D. A. Johnston and P. Plecháč: Equivalence of ferromagnetic spin models on trees and random graphs, J. Phys. A 31 (1998), 475–482.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    M. Krivelevich and B. Sudakov: Coloring random graphs, Information Processing Letters 67 (1998), 71–74.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    F. Krzakala, A. Montanari, F. Ricci-Tersenghi, G. Semerjian and L. Zde- borova: Gibbs states and the set of solutions of random constraint satisfaction prob- lems, Proc. National Academy of Sciences 104 (2007), 10318–10323.MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    F. Krzakala, A. Pagnani and M. Weigt: Threshold values, stability analysis and high-q asymptotics for the coloring problem on random graphs, Phys. Rev. E 70 (2004), 046705.Google Scholar
  29. [29]
    T. Luczak: The chromatic number of random graphs, Combinatorica 11 (1991), 45–54.MathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    T. Luczak: A note on the sharp concentration of the chromatic number of random graphs, Combinatorica 11 (1991), 295–297.MathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    D. Matula: Expose-and-merge exploration and the chromatic number of a random graph, Combinatorica 7 (1987), 275–284.MathSciNetCrossRefMATHGoogle Scholar
  32. [32]
    M. Mézard and A. Montanari: Information, physics and computation, Oxford University Press 2009.Google Scholar
  33. [33]
    M. Mézard and G. Parisi: The Bethe lattice spin glass revisited, Eur. Phys. J. B 20 (2001), 217–233.MathSciNetCrossRefGoogle Scholar
  34. [34]
    M. Mézard and G. Parisi: The cavity method at zero temperature, Journal of Statistical Physics 111 (2003), 1–34.MathSciNetCrossRefMATHGoogle Scholar
  35. [35]
    M. Mézard, M. Palassini and O. Rivoire: Landscape of solutions in constraint satisfaction problems, Phys. Rev. E 95 (2005), 200202.Google Scholar
  36. [36]
    M. Mézard, G. Parisi and R. Zecchina: Analytic and algorithmic solution of random satisfiability problems, Science 297 (2002), 812–815.CrossRefGoogle Scholar
  37. [37]
    M. Molloy: The freezing threshold for k-colourings of a random graph, Proc. 43rd STOC (2012), 921–930.Google Scholar
  38. [38]
    A. Montanari, E. Mossel and A. Sly: The weak limit of Ising models on locally tree-like graphs, Probab. Theory Relat. Fields 152 (2012), 31–51.MathSciNetCrossRefMATHGoogle Scholar
  39. [39]
    A. Montanari, R. Restrepo and P. Tetali: Reconstruction and clustering in random constraint satisfaction problems, SIAM J. Discrete Math. 25 (2011), 771–808.MathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    E. Mossel and A. Sly: Gibbs rapidly samples colorings of G(n;d=n), Probability Theory and Related Fields 148 (2010), 37–69.MathSciNetCrossRefMATHGoogle Scholar
  41. [41]
    E. Shamir and J. Spencer: Sharp concentration of the chromatic number of random graphs G(n;p), Combinatorica 7 (1987), 121–129.MathSciNetCrossRefMATHGoogle Scholar
  42. [42]
    A. Sly: Reconstruction of random colourings, Comm. Math. Phys. 288 (2009), 943–961.MathSciNetCrossRefMATHGoogle Scholar
  43. [43]
    L. Warnke: On the method of typical bounded differences, Combinatorics, Proba- bility and Computing, 25 (2016) 269–299.MathSciNetCrossRefGoogle Scholar
  44. [44]
    L. Zdeborová and F. Krzakala: Phase transitions in the coloring of random graphs, Phys. Rev. E 76 (2007), 031131.Google Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Amin Coja-Oghlan
    • 1
  • Charilaos Efthymiou
    • 1
  • Nor Jaafari
    • 1
  1. 1.Mathematics InstituteGoethe UniversityFrankfurtGermany

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