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Triangle-Free Coloring in LOCAL via Resilient Lovász Local Lemma

Authors Peter Davies-Peck , Xusheng Zhang

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LIPIcs.DISC.2026.29.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Probabilistic algorithms
  • Mathematics of computing → Graph coloring
  • Mathematics of computing → Graph algorithms
Keywords
  • Distributed Graph Coloring
  • Triangle-free Graphs
  • Lovász Local Lemma

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Abstract

The Lovász Local Lemma (LLL) is a probabilistic tool that has been shown to be of central importance in the study of distributed algorithms. For example, the constructive LLL is known to be complete for the class of locally-checkable labeling problems with o(log n) randomized complexities in the LOCAL model. One classic application of the LLL is in coloring graphs with some sparse structure, such as triangle-free graphs. Triangle-free coloring therefore serves as a benchmark problem for techniques for sublogarithmic randomized distributed algorithms. 
The state-of-the-art distributed triangle-free coloring algorithm of Pettie and Su [ICALP 2013, Information and Computation 2015] uses Δ/k colors (where k can be up to (1/4 - ε)ln Δ) and consists of O(k+log^* n) applications of the distributed LLL. However, the distributed LLL is itself a difficult problem; despite significant study, the fastest algorithms known require O(log_Δ n) or O(Δ/logΔ) + log^O(1)log n rounds, and no log^o(1) n-round algorithms exist except for restrictive special cases or graph classes such as trees. These LLL applications therefore dominate the round complexity of the algorithm.
In this work, we adapt the Pettie-Su’s algorithm so that the resulting LLL instances can be solved in log^O(1) log n rounds, by employing the "resilience" definition of Davies [SODA 2023]. This gives an O(k)+log^O(1) log n complexity (since the LLL is not needed when k = log^ω(1) log n), essentially causing the LLL steps to no longer be the bottleneck of the algorithm. As a corollary we obtain the first log^O(1) log n-round algorithms for coloring triangle-free graphs with o(Δ) colors. The same framework also yields a companion girth-5 algorithm, using (1+ε)Δ/ln Δ colors in O(k)+ log^O(1) log n rounds, matching the best known existential upper bound for the number of colors.
The LLL instances we use are among the most difficult for which efficient LOCAL algorithms have been shown, and we hope that this sheds further light on the solvability of the distributed LLL and may lead to further efficient applications.

Cite As Get BibTex

Peter Davies-Peck and Xusheng Zhang. Triangle-Free Coloring in LOCAL via Resilient Lovász Local Lemma. In 40th International Symposium on Distributed Computing (DISC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 397, pp. 29:1-29:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.DISC.2026.29

Author Details

Peter Davies-Peck
  • Durham University, UK
Xusheng Zhang
  • Durham University, UK

Funding

This work is supported by EPSRC New Investigator Award UKRI155 "Distributed Lovász Local Lemma".

References

  1. Miklós Ajtai, János Komlós, and Endre Szemerédi. A note on ramsey numbers. Journal of Combinatorial Theory, Series A, 29(3):354-360, 1980. URL: https://doi.org/10.1016/0097-3165(80)90030-8.
  2. Leonid Barenboim, Michael Elkin, and Fabian Kuhn. Distributed (Δ+1)-coloring in linear (in Δ) time. SIAM Journal on Computing, 43(1):72-95, 2014. URL: https://doi.org/10.1137/12088848X.
  3. Leonid Barenboim, Michael Elkin, Seth Pettie, and Johannes Schneider. The locality of distributed symmetry breaking. In 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science (FOCS), pages 321-330, 2012. URL: https://doi.org/10.1109/FOCS.2012.60.
  4. Anton Bernshteyn. The johansson-molloy theorem for dp-coloring. Random Structures & Algorithms, 54(4):653-664, 2019. URL: https://doi.org/10.1002/rsa.20811.
  5. Béla Bollobás. Chromatic number, girth and maximal degree. Discrete Mathematics, 24(3):311-314, 1978. URL: https://doi.org/10.1016/0012-365X(78)90102-4.
  6. Sebastian Brandt, Orr Fischer, Juho Hirvonen, Barbara Keller, Tuomo Lempiäinen, Joel Rybicki, Jukka Suomela, and Jara Uitto. A lower bound for the distributed lovász local lemma. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, pages 479-488, 2016. URL: https://doi.org/10.1145/2897518.2897570.
  7. Yi-Jun Chang, Qizheng He, Wenzheng Li, Seth Pettie, and Jara Uitto. Distributed edge coloring and a special case of the constructive lovász local lemma. ACM Transactions on Algorithms (TALG), 16(1):1-51, 2019. URL: https://doi.org/10.1145/3365004.
  8. Yi-Jun Chang, Wenzheng Li, and Seth Pettie. Distributed (Δ+1)-coloring via ultrafast graph shattering. SIAM Journal on Computing, 49(3):497-539, 2020. URL: https://doi.org/10.1137/19M1249527.
  9. Yi-Jun Chang and Seth Pettie. A time hierarchy theorem for the LOCAL model. SIAM Journal on Computing, 48(1):33-69, 2019. URL: https://doi.org/10.1137/17M1157957.
  10. Kai-Min Chung, Seth Pettie, and Hsin-Hao Su. Distributed algorithms for the lovász local lemma and graph coloring. Distrib. Comput, 30:261-280, 2017. URL: https://doi.org/10.1007/S00446-016-0287-6.
  11. Peter Davies. Improved distributed algorithms for the lovász local lemma and edge coloring. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 4273-4295, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch163.
  12. Peter Davies-Peck. On the locality of the lovász local lemma. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing, pages 1271-1282, 2025. URL: https://doi.org/10.1145/3717823.3718103.
  13. Peter Davies-Peck and Xusheng Zhang. Triangle-free coloring in local via resilient lovász local lemma, 2026. Full version of this paper. URL: https://arxiv.org/abs/2608.13357.
  14. Manuela Fischer and Mohsen Ghaffari. Sublogarithmic distributed algorithms for lovász local lemma, and the complexity hierarchy. In 31st International Symposium on Distributed Computing (DISC 2017), volume 91 of LIPIcs, pages 18:1-18:16, 2017. URL: https://doi.org/10.4230/LIPIcs.DISC.2017.18.
  15. Mohsen Ghaffari and Christoph Grunau. Near-optimal deterministic network decomposition and ruling set, and improved mis. In 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS), pages 2148-2179, 2024. URL: https://doi.org/10.1109/FOCS61266.2024.00007.
  16. Mohsen Ghaffari, Christoph Grunau, and Václav Rozhoň. Improved deterministic network decomposition. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2211-2231, 2021. URL: https://doi.org/10.1137/1.9781611976465.131.
  17. Mohsen Ghaffari, David G. Harris, and Fabian Kuhn. On derandomizing local distributed algorithms. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 662-673, 2018. URL: https://doi.org/10.1109/FOCS.2018.00069.
  18. David A. Grable and Alessandro Panconesi. Fast distributed algorithms for Brooks-Vizing colorings. Journal of Algorithms, 37(1):85-120, 2000. URL: https://doi.org/10.1006/JAGM.2000.1097.
  19. Magnús M. Halldórsson, Fabian Kuhn, Alexandre Nolin, and Tigran Tonoyan. Near-optimal distributed degree+1 coloring. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 450-463, 2022. URL: https://doi.org/10.1145/3519935.3520023.
  20. Magnús M Halldórsson and Yannic Maus. Distributed delta-coloring under bandwidth limitations. In 38th International Symposium on Distributed Computing (DISC 2024), pages 31:1-31:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.DISC.2024.31.
  21. Magnús M Halldórsson, Yannic Maus, and Alexandre Nolin. Fast distributed vertex splitting with applications. In 36th International Symposium on Distributed Computing (DISC 2022), pages 26:1-26:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.DISC.2022.26.
  22. David G. Harris, Johannes Schneider, and Hsin-Hao Su. Distributed (Δ+1)-coloring in sublogarithmic rounds. Journal of the ACM, 65(4):19:1-19:21, 2018. URL: https://doi.org/10.1145/3178120.
  23. Mohammad Shoaib Jamall. A coloring algorithm for triangle-free graphs, 2011. Preprint. Google Scholar
  24. Öjvind Johansson. Asymptotic choice number for triangle-free graphs. Technical Report 91-5, DIMACS, 1996. Google Scholar
  25. Öjvind Johansson. Simple distributed Δ +1-coloring of graphs. Information Processing Letters, 70(5):229-232, 1999. Google Scholar
  26. Jeong Han Kim. On brooks' theorem for sparse graphs. Combinatorics, Probability and Computing, 4(2):97-132, 1995. URL: https://doi.org/10.1017/S0963548300001638.
  27. A. V. Kostochka. Degree, girth and chromatic number. In Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, volume 18 of Colloq. Math. Soc. János Bolyai, pages 679-696. North-Holland, 1978. Google Scholar
  28. Nathan Linial. Locality in distributed graph algorithms. SIAM Journal on Computing, 21(1):193-201, 1992. URL: https://doi.org/10.1137/0221015.
  29. Michael Molloy. The list chromatic number of graphs with small clique number. Journal of Combinatorial Theory, Series B, 134:264-284, 2019. URL: https://doi.org/10.1016/j.jctb.2018.06.007.
  30. Robin A. Moser and Gábor Tardos. A constructive proof of the general lovász local lemma. Journal of the ACM, 57(2):11:1-11:15, 2010. URL: https://doi.org/10.1145/1667053.1667060.
  31. Seth Pettie and Hsin-Hao Su. Distributed coloring algorithms for triangle-free graphs. Information and Computation, 243:263-280, 2015. URL: https://doi.org/10.1016/j.ic.2014.12.018.
  32. Václav Rozhoň and Mohsen Ghaffari. Polylogarithmic-time deterministic network decomposition and distributed derandomization. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 350-363, 2020. URL: https://doi.org/10.1145/3357713.3384298.
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