Edge-Coloring Sparse Graphs with Δ Colors in Quasilinear Time

Author Łukasz Kowalik



PDF
Thumbnail PDF

File

LIPIcs.ESA.2024.81.pdf
  • Filesize: 1.5 MB
  • 17 pages

Document Identifiers

Author Details

Łukasz Kowalik
  • Institute of Informatics, University of Warsaw, Poland

Acknowledgements

The author wishes to thank Bartłomiej Bosek, Jadwiga Czyżewska, Konrad Majewski and Anna Zych-Pawlewicz for helpful discussions on related topics. The author is also grateful to the reviewers for helpful comments.

Cite AsGet BibTex

Łukasz Kowalik. Edge-Coloring Sparse Graphs with Δ Colors in Quasilinear Time. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 81:1-81:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.81

Abstract

In this paper we show that every graph G of bounded maximum average degree mad(G) and with maximum degree Δ can be edge-colored using the optimal number of Δ colors in quasilinear time, whenever Δ ≥ 2mad(G). The maximum average degree is within a multiplicative constant of other popular graph sparsity parameters like arboricity, degeneracy or maximum density. Our algorithm extends previous results of Chrobak and Nishizeki [Marek Chrobak and Takao Nishizeki, 1990] and Bhattacharya, Costa, Panski and Solomon [Sayan Bhattacharya et al., 2023].

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph coloring
  • Theory of computation → Graph algorithms analysis
Keywords
  • edge coloring
  • algorithm
  • sparse
  • graph
  • quasilinear

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Sepehr Assadi. Faster Vizing and near-Vizing edge coloring algorithms. CoRR, abs/2405.13371, 2024. URL: https://doi.org/10.48550/arXiv.2405.13371.
  2. Anton Bernshteyn. A fast distributed algorithm for (Δ+1)-edge-coloring. J. Comb. Theory, Ser. B, 152:319-352, 2022. URL: https://doi.org/10.1016/J.JCTB.2021.10.004.
  3. Anton Bernshteyn and Abhishek Dhawan. Fast algorithms for Vizing’s theorem on bounded degree graphs. CoRR, abs/2303.05408, 2023. URL: https://doi.org/10.48550/arXiv.2303.05408.
  4. Sayan Bhattacharya, Din Carmon, Martín Costa, Shay Solomon, and Tianyi Zhang. Faster (Δ + 1)-edge coloring: Breaking the m√n time barrier. CoRR, abs/2405.15449, 2024. URL: https://doi.org/10.48550/arXiv.2405.15449.
  5. Sayan Bhattacharya, Martín Costa, Nadav Panski, and Shay Solomon. Density-sensitive algorithms for (Δ+1)-edge coloring. CoRR, abs/2307.02415, 2023. To appear in ESA'24. https://arxiv.org/abs/2307.02415, URL: https://doi.org/10.48550/arXiv.2307.02415.
  6. Sayan Bhattacharya, Martín Costa, Nadav Panski, and Shay Solomon. Arboricity-dependent algorithms for edge coloring. In Hans L. Bodlaender, editor, 19th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2024, June 12-14, 2024, Helsinki, Finland, volume 294 of LIPIcs, pages 12:1-12:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.SWAT.2024.12.
  7. Yan Cao, Guantao Chen, Suyun Jiang, Huiqing Liu, and Fuliang Lu. Average degrees of edge-chromatic critical graphs. Discrete Mathematics, 342(6):1613-1623, 2019. URL: https://doi.org/10.1016/j.disc.2019.02.014.
  8. Aleksander B. G. Christiansen, Eva Rotenberg, and Juliette Vlieghe. Sparsity-parameterised dynamic edge colouring. In Hans L. Bodlaender, editor, 19th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2024, June 12-14, 2024, Helsinki, Finland, volume 294 of LIPIcs, pages 20:1-20:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.SWAT.2024.20.
  9. Aleksander Bjørn Grodt Christiansen. The power of multi-step Vizing chains. In Barna Saha and Rocco A. Servedio, editors, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 1013-1026. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585105.
  10. Marek Chrobak and Takao Nishizeki. Improved edge-coloring algorithms for planar graphs. J. Algorithms, 11(1):102-116, 1990. URL: https://doi.org/10.1016/0196-6774(90)90032-A.
  11. Marek Chrobak and Moti Yung. Fast algorithms for edge-coloring planar graphs. J. Algorithms, 10(1):35-51, 1989. URL: https://doi.org/10.1016/0196-6774(89)90022-9.
  12. Richard Cole and Lukasz Kowalik. New linear-time algorithms for edge-coloring planar graphs. Algorithmica, 50(3):351-368, 2008. URL: https://doi.org/10.1007/S00453-007-9044-3.
  13. Richard Cole, Kirstin Ost, and Stefan Schirra. Edge-coloring bipartite multigraphs in O(E log D) time. Comb., 21(1):5-12, 2001. URL: https://doi.org/10.1007/S004930170002.
  14. Daniel W. Cranston and Douglas B. West. An introduction to the discharging method via graph coloring. Discrete Mathematics, 340(4):766-793, 2017. URL: https://doi.org/10.1016/j.disc.2016.11.022.
  15. Ran Duan, Haoqing He, and Tianyi Zhang. Dynamic edge coloring with improved approximation. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 1937-1945. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.117.
  16. Michael Elkin and Ariel Khuzman. Deterministic simple (1+ε)Δ-edge-coloring in near-linear time, 2024. URL: https://arxiv.org/abs/2401.10538.
  17. Stanley Fiorini. Some remarks on a paper by Vizing on critical graphs. Mathematical Proceedings of the Cambridge Philosophical Society, 77(3):475-483, 1975. URL: https://doi.org/10.1017/S030500410005129X.
  18. Harold N. Gabow, Takao Nishizeki, Oded Kariv, Daniel Leven, and Osamu Terada. Algorithms for edge-coloring graphs. Technical report, Tel Aviv University, 1985. URL: https://www.ecei.tohoku.ac.jp/alg/nishizeki/sub/e/Edge-Coloring.pdf.
  19. Dawit Haile. Bounds on the size of critical edge-chromatic graphs. Ars Comb., 53:85-96, 1999. Google Scholar
  20. Ian Holyer. The NP-completeness of edge-coloring. SIAM J. Comput., 10(4):718-720, 1981. URL: https://doi.org/10.1137/0210055.
  21. Lukasz Kowalik. Approximation scheme for lowest outdegree orientation and graph density measures. In Tetsuo Asano, editor, Algorithms and Computation, 17th International Symposium, ISAAC 2006, Kolkata, India, December 18-20, 2006, Proceedings, volume 4288 of Lecture Notes in Computer Science, pages 557-566. Springer, 2006. URL: https://doi.org/10.1007/11940128_56.
  22. Lukasz Kowalik. Edge-coloring sparse graphs with Δ colors in quasilinear time. CoRR, abs/2401.13839, 2024. URL: https://doi.org/10.48550/arXiv.2401.13839.
  23. Daniel P. Sanders and Yue Zhao. On the size of edge chromatic critical graphs. J. Comb. Theory, Ser. B, 86(2):408-412, 2002. URL: https://doi.org/10.1006/JCTB.2002.2135.
  24. Corwin Sinnamon. Fast and simple edge-coloring algorithms. CoRR, abs/1907.03201, 2019. URL: https://arxiv.org/abs/1907.03201.
  25. Michael Stiebitz, Diego Scheide, Bjarne Toft, and Lene M. Favrholdt. Graph edge coloring: Vizing’s theorem and Goldberg’s conjecture. Wiley Hoboken, NJ, Hoboken, NJ, 2012. Google Scholar
  26. Vadim G. Vizing. On the estimate of the chromatic class of a p-graph. Diskret. Analiz, 3:25-30, 1964. Google Scholar
  27. Vadim G. Vizing. Critical graphs with a given chromatic number. Diskret. Analiz, 5:9-17, 1965. Google Scholar
  28. Vadim G. Vizing. Some unsolved problems in graph theory (in Russian). Uspeki Mat. Nauk., 23:117-134, 1968. English translation in [Russian Mathematical Surveys 23 (1968), 125-141]. Google Scholar
  29. Douglas R. Woodall. The average degree of an edge-chromatic critical graph II. Journal of Graph Theory, 56(3):194-218, 2007. URL: https://doi.org/10.1002/jgt.20259.
  30. Douglas R. Woodall. Erratum: The average degree of an edge-chromatic critical graph II. Journal of Graph Theory, 92(4):488-490, 2019. URL: https://doi.org/10.1002/jgt.22501.
  31. Xiao Zhou, Shin ichi Nakano, and Takao Nishizeki. Edge-coloring partial k-trees. Journal of Algorithms, 21(3):598-617, 1996. URL: https://doi.org/10.1006/jagm.1996.0061.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail