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Sampling Colorings with Fixed Color Class Sizes

Authors Aiya Kuchukova , Will Perkins , Xavier Povill

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LIPIcs.ICALP.2026.134.pdf
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ACM Subject Classification
  • Theory of computation → Generating random combinatorial structures
  • Mathematics of computing → Approximation algorithms
Keywords
  • sampling
  • approximate counting
  • graph coloring
  • zero-freeness
  • Potts model
  • LCLT

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Abstract

In 1970, Hajnal and Szemerédi proved a conjecture of Erdős stating that any graph with maximum degree Δ admits an equitable (Δ+1)-coloring, that is, a coloring where color class sizes differ by at most 1. In 2007 Kierstead and Kostochka reproved their result and provided a polynomial-time algorithm which produces such a coloring. In this paper we study the problem of approximately sampling uniformly random equitable colorings. A series of works gives polynomial-time sampling algorithms for colorings without the color class constraint, the latest improvement being by Carlson and Vigoda for q ≥ 1.809 Δ. In this paper we give a polynomial-time sampling algorithm for equitable colorings when q > 2Δ. Moreover, our results extend to colorings with small deviations from equitable (and as a corollary, establishing their existence). The proof uses the framework of the geometry of polynomials for multivariate polynomials, and as a consequence establishes a multivariate local Central Limit Theorem for color class sizes of uniform random colorings.

Cite As Get BibTex

Aiya Kuchukova, Will Perkins, and Xavier Povill. Sampling Colorings with Fixed Color Class Sizes. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 134:1-134:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.134

Author Details

Aiya Kuchukova
  • School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA
Will Perkins
  • School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA
Xavier Povill
  • Department of Mathematics, Universitat Politècnica de Catalunya, Barcelona, Spain

Funding

  • Kuchukova, Aiya: Supported in part by a Georgia Tech ARC-ACO student fellowship.
  • Perkins, Will: Supported in part by NSF grant CCF-2309708.
  • Povill, Xavier: Supported in part by a travel fellowship from MSCA-RISE-2020 project RandNET (no. 101007705), grant PID2023-147202NB-I00 funded by MICIU/AEI/10.13039/501100011033, and a FPI-UPC grant by Universitat Politècnica de Catalunya and Banco Santander.

References

  1. Alexander Barvinok. Combinatorics and complexity of partition functions, volume 30. Springer, 2016. URL: https://doi.org/10.1007/978-3-319-51829-9.
  2. Ferenc Bencs, Khallil Berrekkal, and Guus Regts. Deterministic approximate counting of colorings with fewer than 2Δ colors via absence of zeros. TheoretiCS, 5, 2026. URL: https://doi.org/10.46298/theoretics.26.1.
  3. Antonio Blanca, Pietro Caputo, Zongchen Chen, Daniel Parisi, Daniel Štefankovič, and Eric Vigoda. On mixing of Markov chains: Coupling, spectral independence, and entropy factorization. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3670-3692. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.145.
  4. Charlie Carlson, Ewan Davies, Alexandra Kolla, and Will Perkins. Computational thresholds for the fixed-magnetization Ising model. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 1459-1472, 2022. URL: https://doi.org/10.1145/3519935.3520003.
  5. Charlie Carlson and Eric Vigoda. Flip dynamics for sampling colorings: Improving (11/6—ε) using a simple metric. In Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2194-2212. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.71.
  6. Sitan Chen, Michelle Delcourt, Ankur Moitra, Guillem Perarnau, and Luke Postle. Improved bounds for randomly sampling colorings via linear programming. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2216-2234. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.134.
  7. Zongchen Chen, Kuikui Liu, Nitya Mani, and Ankur Moitra. Strong spatial mixing for colorings on trees and its algorithmic applications. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 810-845. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00053.
  8. Zongchen Chen, Kuikui Liu, and Eric Vigoda. Spectral independence via stability and applications to Holant-type problems. TheoretiCS, 3, 2024. URL: https://doi.org/10.46298/theoretics.24.16.
  9. Ewan Davies and Will Perkins. Approximately counting independent sets of a given size in bounded-degree graphs. SIAM Journal on Computing, 52(2):618-640, 2023. URL: https://doi.org/10.1137/21m1466220.
  10. Martin Dyer and Alan Frieze. Randomly coloring graphs with lower bounds on girth and maximum degree. Random Structures & Algorithms, 23(2):167-179, 2003. URL: https://doi.org/10.1002/rsa.10087.
  11. Charilaos Efthymiou, Thomas P Hayes, Daniel Štefankovič, and Eric Vigoda. Sampling random colorings of sparse random graphs. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1759-1771. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.115.
  12. András Hajnal and E. Szemerédi. Proof of a conjecture of P. Erdős. Colloq Math Soc János Bolyai, 4, January 1970. Google Scholar
  13. Thomas P Hayes and Alistair Sinclair. A general lower bound for mixing of single-site dynamics on graphs. In 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05), pages 511-520. IEEE, 2005. URL: https://doi.org/10.1109/SFCS.2005.6.
  14. Thomas P Hayes and Eric Vigoda. A non-Markovian coupling for randomly sampling colorings. In 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings., pages 618-627. IEEE, 2003. URL: https://doi.org/10.1109/SFCS.2003.1238234.
  15. Ole J Heilmann and Elliott H Lieb. Theory of monomer-dimer systems. Communications in Mathematical Physics, 25(3):190-232, 1972. URL: https://doi.org/10.1007/BF01877590.
  16. Eoin Hurley, Felix Joos, and Richard Lang. Sufficient conditions for perfect mixed tilings. Journal of Combinatorial Theory, Series B, 170:128-188, 2025. URL: https://doi.org/10.1016/j.jctb.2024.08.007.
  17. Vishesh Jain, Marcus Michelen, Huy Tuan Pham, and Thuy-Duong Vuong. Optimal mixing of the down-up walk on independent sets of a given size. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 1665-1681. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00101.
  18. Vishesh Jain, Will Perkins, Ashwin Sah, and Mehtaab Sawhney. Approximate counting and sampling via local central limit theorems. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 1473-1486, 2022. URL: https://doi.org/10.1145/3519935.3519957.
  19. Mark Jerrum. A very simple algorithm for estimating the number of k-colorings of a low-degree graph. Random Structures & Algorithms, 7(2):157-165, 1995. URL: https://doi.org/10.1002/rsa.3240070205.
  20. Mark Jerrum and Alistair Sinclair. Polynomial-time approximation algorithms for the Ising model. SIAM Journal on computing, 22(5):1087-1116, 1993. URL: https://doi.org/10.1137/0222066.
  21. Mark R Jerrum, Leslie G Valiant, and Vijay V Vazirani. Random generation of combinatorial structures from a uniform distribution. Theoretical computer science, 43:169-188, 1986. URL: https://doi.org/10.1016/0304-3975(86)90174-X.
  22. Hal A Kierstead and Alexandr V Kostochka. A short proof of the Hajnal-Szemerédi theorem on equitable colouring. Combinatorics, Probability and Computing, 17(2):265-270, 2008. URL: https://doi.org/10.1017/S0963548307008619.
  23. Aiya Kuchukova, Marcus Pappik, Will Perkins, and Corrine Yap. Fast and slow mixing of the Kawasaki dynamics on bounded-degree graphs. Random Structures & Algorithms, 67(4):e70038, 2025. URL: https://doi.org/10.1002/rsa.70038.
  24. Aiya Kuchukova, Will Perkins, and Xavier Povill. Sampling colorings with fixed color class sizes, 2026. URL: https://doi.org/10.48550/arXiv.2603.08259.
  25. Jingcheng Liu, Alistair Sinclair, and Piyush Srivastava. Correlation decay and partition function zeros: Algorithms and phase transitions. SIAM J. Comput., 54(4):S19-200, 2025. URL: https://doi.org/10.1137/20m1317384.
  26. Kuikui Liu. From coupling to spectral independence and blackbox comparison with the down-up walk. In Mary Wootters and Laura Sanità, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021), volume 207 of Leibniz International Proceedings in Informatics (LIPIcs), pages 32:1-32:21, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.32.
  27. Fabio Martinelli, Alistair Sinclair, and Dror Weitz. Fast mixing for independent sets, colorings, and other models on trees. Random Structures & Algorithms, 31(2):134-172, 2007. URL: https://doi.org/10.1002/rsa.20132.
  28. Marc Mézard and Giorgio Parisi. Mean-field theory of randomly frustrated systems with finite connectivity. EPL (Europhysics Letters), 3(10):1067-1074, 1987. URL: https://doi.org/10.1209/0295-5075/3/10/002.
  29. Marcus Michelen and Julian Sahasrabudhe. Central limit theorems and the geometry of polynomials. Journal of the European Mathematical Society, 2024. Google Scholar
  30. Michael Molloy. The Glauber dynamics on colourings of a graph with high girth and maximum degree. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 91-98, 2002. URL: https://doi.org/10.1145/509907.509924.
  31. Elchanan Mossel and Allan Sly. Gibbs rapidly samples colorings of G(n, d/n). Probability theory and related fields, 148(1):37-69, 2010. URL: https://doi.org/10.1007/s00440-009-0222-x.
  32. Viresh Patel, Guus Regts, and Ayla Stam. A near-optimal zero-free disk for the Ising model. Combinatorial Theory, 4(2), 2024. URL: https://doi.org/10.5070/c64264237.
  33. Han Peters and Guus Regts. On a conjecture of Sokal concerning roots of the independence polynomial. Michigan Mathematical Journal, 68(1):33-55, 2019. Google Scholar
  34. Han Peters and Guus Regts. Location of zeros for the partition function of the Ising model on bounded degree graphs. Journal of the London Mathematical Society, 101(2):765-785, 2020. URL: https://doi.org/10.1112/jlms.12286.
  35. Dana Randall and David Wilson. Sampling spin configurations of an Ising system. In Symposium on Discrete Algorithms: Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms, volume 17, pages 959-960, 1999. URL: http://dl.acm.org/citation.cfm?id=314500.314945.
  36. Guus Regts. Absence of zeros implies strong spatial mixing. Probability Theory and Related Fields, 186(1):621-641, 2023. URL: https://doi.org/10.1007/s00440-023-01190-z.
  37. Allan Sly and Yumeng Zhang. The Glauber dynamics of colorings on trees is rapidly mixing throughout the nonreconstruction regime. Annals of Applied Probability, 27(5):2646-2674, 2017. Google Scholar
  38. Eric Vigoda. Improved bounds for sampling colorings. In 40th Annual Symposium on Foundations of Computer Science (Cat. No. 99CB37039), pages 51-59. IEEE, 1999. URL: https://doi.org/10.1109/SFFCS.1999.814577.
  39. Chen-Ning Yang and Tsung-Dao Lee. Statistical theory of equations of state and phase transitions. i. Theory of condensation. Physical Review, 87(3):404, 1952. URL: https://doi.org/10.1103/PhysRev.87.404.
  40. Penghui Yao, Yitong Yin, and Xinyuan Zhang. Polynomial-time approximation of zero-free partition functions. In Mikolaj Bojanczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, Paris, France, July 4-8, 2022, volume 229 of LIPIcs, pages 108:1-108:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.108.
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