Skip to main content
Log in

Lower Bounds on the Chromatic Number of Random Graphs

  • Original paper
  • Published:
Combinatorica Aims and scope Submit manuscript

Abstract

We prove that a formula predicted on the basis of non-rigorous physics arguments [Zdeborová and Krzakala: Phys. Rev. E (2007)] provides a lower bound on the chromatic number of sparse random graphs. The proof is based on the interpolation method from mathematical physics. In the case of random regular graphs the lower bound can be expressed algebraically, while in the case of the binomial random we obtain a variational formula. As an application we calculate improved explicit lower bounds on the chromatic number of random graphs for small (average) degrees. Additionally, we show how asymptotic formulas for large degrees that were previously obtained by lengthy and complicated combinatorial arguments can be re-derived easily from these new results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. Achlioptas and E. Friedgut: A sharp threshold for k-colorability, Random Struct. Algorithms 14 (1999), 63–70.

    Article  MathSciNet  MATH  Google Scholar 

  2. D. Achlioptas and C. Moore: Almost all graphs with average degree 4 are 3-colorable, Journal of Computer and System Sciences 67 (2003), 441–471.

    Article  MathSciNet  MATH  Google Scholar 

  3. D. Achlioptas and C. Moore: The chromatic number of random regular graphs, Proc. 8th RANDOM (2004), 219–228.

  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.

    Article  MathSciNet  MATH  Google Scholar 

  5. N. Alon and M. Krivelevich: The concentration of the chromatic number of random graphs, Combinatorica 17 (1997), 303–313

    Article  MathSciNet  MATH  Google Scholar 

  6. V. Bapst, A. Coja-Oghlan, S. Hetterich, F. Rassmann and D. Vilenchik: The condensation phase transition in random graph coloring, Communications in Mathematical Physics 341 (2016), 543–606.

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Bayati, D. Gamarnik and P. Tetali: Combinatorial approach to the interpolation method and scaling limits in sparse random graphs, Annals of Probability 41 (2013), 4080–4115.

    Article  MathSciNet  MATH  Google Scholar 

  8. B. Bollobás: The chromatic number of random graphs, Combinatorica 8 (1988), 49–55

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Coja-Oghlan: Upper-bounding the k-colorability threshold by counting covers, Electronic Journal of Combinatorics 20 (2013), P32.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Coja-Oghlan, C. Efthymiou and S. Hetterich: On the chromatic number of random regular graphs, Journal of Combinatorial Theory, Series B 116 (2016), 367–439.

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Coja-Oghlan, A. Ergür, P. Gao, S. Hetterich and M. Rolvien: The rank of sparse random matrices, Proc. 31st SODA (2020), 579–591.

  12. A. Coja-Oghlan, F. Krzakala, W. Perkins and L. Zdeborova: Information-theoretic thresholds from the cavity method, Advances in Mathematics 333 (2018), 694–795.

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Coja-Oghlan and K. Panagiotou: The asymptotic k-SAT threshold, Advances in Mathematics 288 (2016), 985–1068.

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Coja-Oghlan, K. Panagiotou and A. Steger: On the chromatic number of random graphs, Journal of Combinatorial Theory, Series B 98 (2008), 980–993.

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Coja-Oghlan and W. Perkins: Spin systems on Bethe lattices, Communications in Mathematical Physics 372 (2019), 441–523.

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Coja-Oghlan and D. Vilenchik: The chromatic number of random graphs for most average degrees, International Mathematics Research Notices 2016 (2016), 5801–5859.

    Article  MathSciNet  MATH  Google Scholar 

  17. C. Cooper, A. Frieze, B. Reed and O. Riordan: Random regular graphs of non-constant degree: independence and chromatic number, Comb. Probab. Comput. 11 (2002), 323–341.

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Diaz, A. Kaporis, G. Kemkes, L. Kirousis, X. Pérez and N. Wormald: On the chromatic number of a random 5-regular graph, Journal of Graph Theory 61 (2009), 157–191.

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Ding, A. Sly and N. Sun: Satisfiability threshold for random regular NAE-SAT, Communications in Mathematical Physics 341 (2016), 435–489.

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Ding, A. Sly and N. Sun: Maximum independent sets on random regular graphs, Acta Math. 217 (2016), 263–340.

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Ding, A. Sly and N. Sun: Proof of the satisfiability conjecture for large k, Proc. 47th STOC (2015), 59–68.

  22. O. Dubois and J. Mandler: On the non-3-colourability of random graphs, arXiv:math/0209087, 2002.

  23. P. Erdős and A. Rényi: On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 (1960), 17–61.

    MathSciNet  MATH  Google Scholar 

  24. S. Franz and M. Leone: Replica bounds for optimization problems and diluted spin systems, J. Stat. Phys. 111 (2003), 535–564.

    Article  MathSciNet  MATH  Google Scholar 

  25. A. Frieze and T. Łuczak: On the independence and chromatic numbers of random regular graphs, J. Comb. Theory B 54 (1992), 123–132.

    Article  MathSciNet  MATH  Google Scholar 

  26. S. Janson, T. Łuczak and A. Ruciński: Random Graphs, Wiley, 2000.

  27. F. Guerra: Broken replica symmetry bounds in the mean field spin glass model, Comm. Math. Phys., 233 (2003), 1–12.

    Article  MathSciNet  MATH  Google Scholar 

  28. G. Kemkes, X. Pérez-Giménez and N. Wormald: On the chromatic number of random d-regular graphs, Advances in Mathematics 223 (2010), 300–328.

    Article  MathSciNet  MATH  Google Scholar 

  29. M. Krivelevich, B. Sudakov, V. Vu and N. Wormald: Random regular graphs of high degree, Random Struct. Algor. 18 (2001), 346–363.

    Article  MathSciNet  MATH  Google Scholar 

  30. M. Lelarge and M. Oulamara: Replica bounds by combinatorial interpolation for diluted spin systems, J. Stat. Phys 173 (2018), 917–940.

    Article  MathSciNet  MATH  Google Scholar 

  31. T. Łuczak: The chromatic number of random graphs, Combinatorica 11 (1991), 45–54

    Article  MathSciNet  MATH  Google Scholar 

  32. D. Matula: Expose-and-merge exploration and the chromatic number of a random graph, Combinatorica 7 (1987), 275–284.

    Article  MathSciNet  MATH  Google Scholar 

  33. D. Matula and L. Kučera: An expose-and-merge algorithm and the chromatic number of a random graph. Proc. Random Graphs 87 (1987), 175–187.

    MATH  Google Scholar 

  34. M. Mézard and A. Montanari: Information, Physics and Computation, Oxford University Press, 2009.

  35. M. Mézard and G. Parisi: The Bethe lattice spin glass revisited, European Physical Journal B 20 (2001), 217–233.

    Article  MathSciNet  Google Scholar 

  36. D. Panchenko: The Sherrington-Kirkpatrick model, Springer, 2013.

  37. D. Panchenko: Spin glass models from the point of view of spin distributions, Annals of Probability 41 (2013), 1315–1361.

    Article  MathSciNet  MATH  Google Scholar 

  38. D. Panchenko and M. Talagrand: Bounds for diluted mean-fields spin glass models, Probab. Theory Relat. Fields 130 (2004), 319–336.

    Article  MathSciNet  MATH  Google Scholar 

  39. E. Shamir and J. Spencer: Sharp concentration of the chromatic number of random graphs Gn,p, Combinatorica 7 (1987), 121–129.

    Article  MathSciNet  MATH  Google Scholar 

  40. L. Shi and N. Wormald: Colouring random 4-regular graphs, Combinatorics, Probability and Computing 16 (2007), 309–344.

    Article  MathSciNet  MATH  Google Scholar 

  41. L. Shi and N. Wormald: Colouring random regular graphs, Combinatorics, Probability and Computing 16 (2007), 459–494.

    Article  MathSciNet  MATH  Google Scholar 

  42. A. Sly, N. Sun and Y. Zhang: The number of solutions for random regular NAE-SAT, Proc. 57th FOCS (2016), 724–731; full version available as arXiv:1604.08546.

  43. M. Talagrand: Spin glasses: a challenge for mathematicians, Springer, 2003.

  44. L. Zdeborová and F. Krzakala: Phase transitions in the coloring of random graphs, Phys. Rev. E 76 (2007), 031131.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgment

We thank Viktor Harangi and an anonymous reviewer for their very careful reading of our manuscript and their extremely accurate and helpful comments, which led to several improvements and corrections.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Amin Coja-Oghlan.

Additional information

Research of the first author supported in part by DFG CO 646. Research of the third author supported by the Australian Research Council Discovery Project DP190100977.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ayre, P., Coja-Oghlan, A. & Greenhill, C. Lower Bounds on the Chromatic Number of Random Graphs. Combinatorica 42, 617–658 (2022). https://doi.org/10.1007/s00493-021-4236-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00493-021-4236-z

Mathematics Subject Classification (2010)

Navigation