Document Open Access Logo

Faster 3-Colouring Algorithm for Graphs of Diameter 3

Authors Carla Groenland , Hidde Koerts , Sophie Spirkl

PDF

File

Thumbnail PDF
PDF
LIPIcs.WG.2026.19.pdf
  • Filesize: 0.86 MB
  • 13 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Mathematics of computing → Graph coloring
Keywords
  • 3-colouring
  • subexponential-time algorithm
  • diameter-3 graphs

Metrics

Abstract

We show that given an n-vertex graph G of diameter 3 we can decide if G is 3-colourable in time 2^{O(n^{2/3-ε})} for any ε < 1/33. This improves on the previous best algorithm of 2^{O((nlog n)^{2/3})} from Dębski, Piecyk and Rzążewski [Faster 3-coloring of small-diameter graphs, ESA 2021].

Cite As Get BibTex

Carla Groenland, Hidde Koerts, and Sophie Spirkl. Faster 3-Colouring Algorithm for Graphs of Diameter 3. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 19:1-19:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.WG.2026.19

Author Details

Carla Groenland
  • Faculty of Electrical Engineering, Mathematics and Computer Science, Technical University Delft, The Netherlands
Hidde Koerts
  • Department of Combinatorics and Optimization, University of Waterloo, Canada
Sophie Spirkl
  • Department of Combinatorics and Optimization, University of Waterloo, Canada

Funding

  • Groenland, Carla: Supported by the Dutch Research Council (NWO, VI.Veni.232.073).
  • Spirkl, Sophie: We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), [funding reference number RGPIN-2020-03912]. Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG), [numéro de référence RGPIN-2020-03912]. This project was funded in part by the Government of Ontario. This research was conducted while Spirkl was an Alfred P. Sloan Fellow.

References

  1. Marthe Bonamy, Konrad K. Dabrowski, Carl Feghali, Matthew Johnson, and Daniël Paulusma. Independent feedback vertex sets for graphs of bounded diameter. Information Processing Letters, 131:26-32, 2018. URL: https://doi.org/10.1016/j.ipl.2017.11.004.
  2. Flavia Bonomo, Maria Chudnovsky, Peter Maceli, Oliver Schaudt, Maya Stein, and Mingxian Zhong. Three-coloring and list three-coloring of graphs without induced paths on seven vertices. Combinatorica, 38(4):779-801, 2018. URL: https://doi.org/10.1007/s00493-017-3553-8.
  3. Valentin Bouquet, François Delbot, Christophe Picouleau, and Stéphane Rovedakis. On the complexity of Dominating Set for graphs with fixed diameter. Theoretical Computer Science, 1001(C), 2024. URL: https://doi.org/10.1016/j.tcs.2024.114561.
  4. Christoph Brause, Petr Golovach, Barnaby Martin, Pascal Ochem, Daniël Paulusma, and Siani Smith. Acyclic, star, and injective colouring: bounding the diameter. The Electronic Journal of Combinatorics, pages P2-43, 2022. URL: https://doi.org/10.37236/10738.
  5. Christoph Brause, Petr Golovach, Barnaby Martin, Daniël Paulusma, and Siani Smith. Partitioning H-free graphs of bounded diameter. Theoretical Computer Science, 930(C):37-52, 2022. URL: https://doi.org/10.1016/j.tcs.2022.07.009.
  6. Victor A Campos, Guilherme CM Gomes, Allen Ibiapina, Raul Lopes, Ignasi Sau, and Ana Silva. Coloring problems on bipartite graphs of small diameter. The Electronic Journal of Combinatorics, 28(2):P2.14, 2021. URL: https://doi.org/10.37236/9931.
  7. Maria Chudnovsky, Sophie Spirkl, and Mingxian Zhong. Four-coloring P₆-free graphs, pages 1239-1256. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.76.
  8. Michał Dębski, Marta Piecyk, and Paweł Rzążewski. Faster 3-coloring of small-diameter graphs. SIAM Journal on Discrete Mathematics, 36(3):2205-2224, 2022. URL: https://doi.org/10.1137/21M1447714.
  9. Keith Edwards. The complexity of colouring problems on dense graphs. Theoretical Computer Science, 43:337-343, 1986. URL: https://doi.org/10.1016/0304-3975(86)90184-2.
  10. C. Groenland, K. Okrasa, P. Rzążewski, A.D. Scott, P. Seymour, and S. Spirkl. H-colouring P_t-free graphs in subexponential time. Discrete Applied Mathematics, 267:184-189, 2019. URL: https://doi.org/10.1016/j.dam.2019.04.010.
  11. C.T. Hoàng, M. Kamiński, V. Lozin, J. Sawada, and X. Shu. Deciding k-colorability of P₅-free graphs in polynomial time. Algorithmica, 57(1):74-81, 2010. URL: https://doi.org/10.1007/s00453-008-9197-8.
  12. Ian Holyer. The NP-completeness of edge coloring. SIAM Journal on Computing, 10(4):718-720, 1981. URL: https://doi.org/10.1137/0210055.
  13. S. Huang. Improved complexity results on k-coloring P_t-free graphs. European Journal of Combinatorics, 51:336-346, 2016. URL: https://doi.org/10.1016/j.ejc.2015.06.005.
  14. D. Leven and Z. Galil. NP-completeness of finding the chromatic index of regular graphs. Journal of Algorithms, 4(1):35-44, 1983. URL: https://doi.org/10.1016/0196-6774(83)90032-9.
  15. Barnaby Martin, Daniël Paulusma, and Siani Smith. Colouring H-free graphs of bounded diameter. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019), volume 138, pages 14:1-14:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.MFCS.2019.14.
  16. Barnaby Martin, Daniël Paulusma, and Siani Smith. Colouring generalized claw-free graphs and graphs of large girth: Bounding the diameter. Theoretical Computer Science, 931:104-116, 2022. URL: https://doi.org/10.1016/j.tcs.2022.07.034.
  17. Barnaby Martin, Daniël Paulusma, and Siani Smith. Colouring graphs of bounded diameter in the absence of small cycles. Discrete Applied Mathematics, 314:150-161, 2022. URL: https://doi.org/10.1016/j.dam.2022.02.026.
  18. George B. Mertzios and Paul G. Spirakis. Algorithms and almost tight results for 3-colorability of small diameter graphs. Algorithmica, 74(1):385-414, 2016. URL: https://doi.org/10.1007/s00453-014-9949-6.
  19. M. Mitzenmacher and E. Upfal. Probability and Computing. Cambridge University Press, 2017. URL: https://doi.org/10.1017/CBO9780511813603.
  20. Daniël Paulusma. Open problems on graph coloring for special graph classes. In Ernst W. Mayr, editor, Graph-Theoretic Concepts in Computer Science (WG), pages 16-30, Berlin, Heidelberg, 2016. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-662-53174-7_2.
  21. Marta Piecyk. C_2k+ 1-Coloring of bounded-diameter graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024), volume 306 of Leibniz International Proceedings in Informatics (LIPIcs), pages 78:1-78:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.MFCS.2024.78.
  22. Marcin Pilipczuk, Michał Pilipczuk, and Paweł Rzążewski. Quasi-polynomial-time algorithm for Independent Set in P_t-free graphs via shrinking the space of induced paths, pages 204-209. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976472.23.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact