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.
Similar content being viewed by others
References
D. Achlioptas and E. Friedgut: A sharp threshold for k-colorability, Random Struct. Algorithms 14 (1999), 63–70.
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.
D. Achlioptas and C. Moore: The chromatic number of random regular graphs, Proc. 8th RANDOM (2004), 219–228.
D. Achlioptas and A. Naor: The two possible values of the chromatic number of a random graph, Annals of Mathematics 162 (2005), 1333–1349.
N. Alon and M. Krivelevich: The concentration of the chromatic number of random graphs, Combinatorica 17 (1997), 303–313
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.
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.
B. Bollobás: The chromatic number of random graphs, Combinatorica 8 (1988), 49–55
A. Coja-Oghlan: Upper-bounding the k-colorability threshold by counting covers, Electronic Journal of Combinatorics 20 (2013), P32.
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.
A. Coja-Oghlan, A. Ergür, P. Gao, S. Hetterich and M. Rolvien: The rank of sparse random matrices, Proc. 31st SODA (2020), 579–591.
A. Coja-Oghlan, F. Krzakala, W. Perkins and L. Zdeborova: Information-theoretic thresholds from the cavity method, Advances in Mathematics 333 (2018), 694–795.
A. Coja-Oghlan and K. Panagiotou: The asymptotic k-SAT threshold, Advances in Mathematics 288 (2016), 985–1068.
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.
A. Coja-Oghlan and W. Perkins: Spin systems on Bethe lattices, Communications in Mathematical Physics 372 (2019), 441–523.
A. Coja-Oghlan and D. Vilenchik: The chromatic number of random graphs for most average degrees, International Mathematics Research Notices 2016 (2016), 5801–5859.
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.
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.
J. Ding, A. Sly and N. Sun: Satisfiability threshold for random regular NAE-SAT, Communications in Mathematical Physics 341 (2016), 435–489.
J. Ding, A. Sly and N. Sun: Maximum independent sets on random regular graphs, Acta Math. 217 (2016), 263–340.
J. Ding, A. Sly and N. Sun: Proof of the satisfiability conjecture for large k, Proc. 47th STOC (2015), 59–68.
O. Dubois and J. Mandler: On the non-3-colourability of random graphs, arXiv:math/0209087, 2002.
P. Erdős and A. Rényi: On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 (1960), 17–61.
S. Franz and M. Leone: Replica bounds for optimization problems and diluted spin systems, J. Stat. Phys. 111 (2003), 535–564.
A. Frieze and T. Łuczak: On the independence and chromatic numbers of random regular graphs, J. Comb. Theory B 54 (1992), 123–132.
S. Janson, T. Łuczak and A. Ruciński: Random Graphs, Wiley, 2000.
F. Guerra: Broken replica symmetry bounds in the mean field spin glass model, Comm. Math. Phys., 233 (2003), 1–12.
G. Kemkes, X. Pérez-Giménez and N. Wormald: On the chromatic number of random d-regular graphs, Advances in Mathematics 223 (2010), 300–328.
M. Krivelevich, B. Sudakov, V. Vu and N. Wormald: Random regular graphs of high degree, Random Struct. Algor. 18 (2001), 346–363.
M. Lelarge and M. Oulamara: Replica bounds by combinatorial interpolation for diluted spin systems, J. Stat. Phys 173 (2018), 917–940.
T. Łuczak: The chromatic number of random graphs, Combinatorica 11 (1991), 45–54
D. Matula: Expose-and-merge exploration and the chromatic number of a random graph, Combinatorica 7 (1987), 275–284.
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.
M. Mézard and A. Montanari: Information, Physics and Computation, Oxford University Press, 2009.
M. Mézard and G. Parisi: The Bethe lattice spin glass revisited, European Physical Journal B 20 (2001), 217–233.
D. Panchenko: The Sherrington-Kirkpatrick model, Springer, 2013.
D. Panchenko: Spin glass models from the point of view of spin distributions, Annals of Probability 41 (2013), 1315–1361.
D. Panchenko and M. Talagrand: Bounds for diluted mean-fields spin glass models, Probab. Theory Relat. Fields 130 (2004), 319–336.
E. Shamir and J. Spencer: Sharp concentration of the chromatic number of random graphs Gn,p, Combinatorica 7 (1987), 121–129.
L. Shi and N. Wormald: Colouring random 4-regular graphs, Combinatorics, Probability and Computing 16 (2007), 309–344.
L. Shi and N. Wormald: Colouring random regular graphs, Combinatorics, Probability and Computing 16 (2007), 459–494.
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.
M. Talagrand: Spin glasses: a challenge for mathematicians, Springer, 2003.
L. Zdeborová and F. Krzakala: Phase transitions in the coloring of random graphs, Phys. Rev. E 76 (2007), 031131.
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
Corresponding author
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
About this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-021-4236-z