Document Open Access Logo

Polychromatic 2-Colorings with Bounded Discrepancy for Triangulations

Authors Alma Arevalo Loyola , Ahmad Biniaz , Prosenjit Bose , Thomas Shermer

PDF

File

Thumbnail PDF
PDF
LIPIcs.SWAT.2026.33.pdf
  • Filesize: 0.78 MB
  • 13 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph coloring
  • Theory of computation → Computational geometry
Keywords
  • polychromatic coloring
  • triangulation
  • balanced coloring
  • matching

Metrics

Abstract

A polychromatic 2-coloring of a triangulation is a 2-coloring of the vertices such that no face is monochromatic. The discrepancy of a coloring is the maximum difference between the sizes of the color classes. Asayama and Matsumoto (Graphs and Combinatorics, 2022) proved that every triangulation admits a polychromatic 2-coloring with discrepancy at most (5n-16)/9, and that there exists a class of triangulations for which every polychromatic 2-coloring has discrepancy at least n/3 - 2, where n is the number of vertices. We improve the upper bound, showing that every triangulation admits a polychromatic 2-coloring with discrepancy at most (3n-16)/7 and such a 2-coloring can be computed in quadratic time. We also show a discrepancy of at most n-4M/3 for triangulations with a matching of size M. This implies, for example, that Delaunay triangulations admit a discrepancy of at most n/3. We provide a linear-time algorithm to compute a 2-coloring whose discrepancy is at most (5n-24)/7.

Cite As Get BibTex

Alma Arevalo Loyola, Ahmad Biniaz, Prosenjit Bose, and Thomas Shermer. Polychromatic 2-Colorings with Bounded Discrepancy for Triangulations. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 33:1-33:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SWAT.2026.33

Author Details

Alma Arevalo Loyola
  • School of Computer Science, Carleton University, Ottawa, Canada
Ahmad Biniaz
  • School of Computer Science, University of Windsor, Canada
Prosenjit Bose
  • School of Computer Science, Carleton University, Ottawa, Canada
Thomas Shermer
  • School of Computing Science, Simon Fraser University, Burnaby, Canada

Funding

  • Biniaz, Ahmad: Supported in part by NSERC.
  • Bose, Prosenjit: Supported in part by NSERC.
  • Shermer, Thomas: Supported in part by NSERC.

Acknowledgements

Part of this work was done at the 12th Annual Workshop on Geometry and Graphs, held at Bellairs Research Institute in Barbados in February 2025. We thank the organizers and the participants.

References

  1. Yoshihiro Asayama and Naoki Matsumoto. Balanced polychromatic 2-coloring of triangulations. Graphs and Combinatorics, 38(1):1-12, 2022. Google Scholar
  2. Therese C. Biedl, Prosenjit Bose, Erik D. Demaine, and Anna Lubiw. Efficient algorithms for Petersen’s matching theorem. J. Algorithms, 38(1):110-134, 2001. URL: https://doi.org/10.1006/JAGM.2000.1132.
  3. Ahmad Biniaz. A short proof of the toughness of delaunay triangulations. J. Comput. Geom., 12(1):35-39, 2021. Also in SOSA'20. URL: https://doi.org/10.20382/JOCG.V12I1A2.
  4. J.A. Bondy and U.S.R Murty. Graph Theory. Springer Publishing Company, Incorporated, 1st edition, 2008. Google Scholar
  5. Prosenjit Bose, David Kirkpatrick, and Zaiqing Li. Worst-case-optimal algorithms for guarding planar graphs and polyhedral surfaces. Computational Geometry, 26(3):209-219, 2003. URL: https://doi.org/10.1016/S0925-7721(03)00027-0.
  6. Prosenjit Bose, Thomas Shermer, Godfried Toussaint, and Binhai Zhu. Guarding polyhedral terrains. Computational Geometry, 7(3):173-185, 1997. URL: https://doi.org/10.1016/0925-7721(95)00034-8.
  7. Yair Caro and Yehuda Roditty. On the vertex-independence number and star decomposition of graphs. Ars Combin, 20:167-180, 1985. Google Scholar
  8. Július Czap and Stanislav Jendrol. Facially-constrained colorings of plane graphs: A survey. Discrete Mathematics, 340(11):2691-2703, 2017. URL: https://doi.org/10.1016/J.DISC.2016.07.026.
  9. Reinhard Diestel. Graph Theory, volume 173 of Graduate Texts in Mathematics. Springer-Verlag, Heidelberg, 6 edition, 2025. Google Scholar
  10. M.B. Dillencourt. Toughness and delaunay triangulations. Discrete & computational geometry, 5(6):575-602, 1990. URL: https://doi.org/10.1007/BF02187810.
  11. Tommy R Jensen and Bjarne Toft. Graph coloring problems. John Wiley & Sons, 2011. Google Scholar
  12. André Kündgen and Carsten Thomassen. Spanning quadrangulations of triangulated surfaces. In Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, volume 87, pages 357-368. Springer, 2017. Google Scholar
  13. Walter Meyer. Equitable coloring. The American mathematical monthly, 80(8):920-922, 1973. Google Scholar
  14. Takao Nishizeki and Ilker Baybars. Lower bounds on the cardinality of the maximum matchings of planar graphs. Discrete Mathematics, 28(3):255-267, 1979. URL: https://doi.org/10.1016/0012-365X(79)90133-X.
  15. Neil Robertson, Daniel Sanders, Paul Seymour, and Robin Thomas. The four-colour theorem. Journal of Combinatorial Theory, Series B, 70(1):2-44, 1997. URL: https://doi.org/10.1006/JCTB.1997.1750.
  16. Tait. 4. on the colouring of maps. Proceedings of the Royal Society of Edinburgh, 10:501-503, 1880. URL: https://doi.org/10.1017/S0370164600044229.
  17. Zsolt Tuza. Graph coloring. In Handbook of Graph Theory, chapter 5, pages 408-438. CRC Press, 2nd edition, November 2013. Google Scholar
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