Document Open Access Logo

Proper Coloring of Geometric Hypergraphs

Authors Balázs Keszegh, Dömötör Pálvölgyi

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2017.47.pdf
  • Filesize: 0.84 MB
  • 15 pages

Document Identifiers

Author Details

Balázs Keszegh
Dömötör Pálvölgyi

Cite AsGet BibTex

Balázs Keszegh and Dömötör Pálvölgyi. Proper Coloring of Geometric Hypergraphs. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 47:1-47:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.SoCG.2017.47

Abstract

We study whether for a given planar family F there is an m such that any finite set of points can be 3-colored such that any member of F that contains at least m points contains two points with different colors. We conjecture that if F is a family of pseudo-disks, then m=3 is sufficient. We prove that when F is the family of all homothetic copies of a given convex polygon, then such an m exists. We also study the problem in higher dimensions.
Keywords
  • discrete geometry
  • decomposition of multiple coverings
  • geometric hypergraph coloring

Metrics

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

References

  1. Eyal Ackerman, Balázs Keszegh, and Máté Vizer. Coloring points with respect to squares. In Sándor P. Fekete and Anna Lubiw, editors, 32nd International Symposium on Computational Geometry, SoCG 2016, June 14-18, 2016, Boston, MA, USA, volume 51 of LIPIcs, pages 5:1-5:16. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  2. Greg Aloupis, Jean Cardinal, Sébastien Collette, Stefan Langerman, and Shakhar Smorodinsky. Coloring geometric range spaces. Discrete & Computational Geometry, 41(2):348-362, 2009. Google Scholar
  3. Andrei Asinowski, Jean Cardinal, Nathann Cohen, Sébastien Collette, Thomas Hackl, Michael Hoffmann, Kolja B. Knauer, Stefan Langerman, Michal Lason, Piotr Micek, Günter Rote, and Torsten Ueckerdt. Coloring hypergraphs induced by dynamic point sets and bottomless rectangles. In Frank Dehne, Roberto Solis-Oba, and Jörg-Rüdiger Sack, editors, Algorithms and Data Structures - 13th International Symposium, WADS 2013, London, ON, Canada, August 12-14, 2013. Proceedings, volume 8037 of Lecture Notes in Computer Science, pages 73-84. Springer, 2013. Google Scholar
  4. Prosenjit Bose, Paz Carmi, Sébastien Collette, and Michiel H. M. Smid. On the stretch factor of convex delaunay graphs. J. of Computational Geometry, 1(1):41-56, 2010. Google Scholar
  5. Peter Brass, William O. J. Moser, and János Pach. Research problems in discrete geometry. Springer, 2005. Google Scholar
  6. Sarit Buzaglo, Rom Pinchasi, and Günter Rote. Topological hypergraphs. In János Pach, editor, Thirty Essays on Geometric Graph Theory, pages 71-81. Springer New York, 2013. URL: http://dx.doi.org/10.1007/978-1-4614-0110-0_6.
  7. Jean Cardinal, Kolja Knauer, Piotr Micek, and Torsten Ueckerdt. Making triangles colorful. J. of Computational Geometry, 4:240-246, 2013. Google Scholar
  8. Jean Cardinal, Kolja Knauer, Piotr Micek, and Torsten Ueckerdt. Making octants colorful and related covering decomposition problems. SIAM J. on Discrete Math., 28(4):1948-1959, 2014. Google Scholar
  9. Jean Cardinal and Matias Korman. Coloring planar homothets and three-dimensional hypergraphs. Computational Geometry, 46(9):1027-1035, 2013. Google Scholar
  10. Xiaomin Chen, János Pach, Mario Szegedy, and Gábor Tardos. Delaunay graphs of point sets in the plane with respect to axis-parallel rectangles. Random Struct. Algorithms, 34(1):11-23, 2009. URL: http://dx.doi.org/10.1002/rsa.20246.
  11. Louis Esperet and Gwenaël Joret. Colouring planar graphs with three colours and no large monochromatic components. Combinatorics, Probability & Computing, 23(4):551-570, 2014. Google Scholar
  12. Radoslav Fulek. Coloring geometric hypergraph defined by an arrangement of half-planes. In Proceedings of the 22nd Annual Canadian Conference on Computational Geometry, Winnipeg, Manitoba, Canada, August 9-11, 2010, pages 71-74, 2010. Google Scholar
  13. Matt Gibson and Kasturi R. Varadarajan. Decomposing coverings and the planar sensor cover problem. In 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, October 25-27, 2009, Atlanta, Georgia, USA, pages 159-168. IEEE Computer Society, 2009. URL: http://dx.doi.org/10.1109/FOCS.2009.54.
  14. Wayne Goddard. Acyclic colorings of planar graphs. Discrete Math., 91(1):91-94, 1991. Google Scholar
  15. B. Gonska and A. Padrol. Neighborly inscribed polytopes and delaunay triangulations. Advances in Geometry, 16(3):349-360, 2016. Google Scholar
  16. A. W. Hales and R. I. Jewett. Regularity and positional games. Trans. Amer. Math. Soc., 106:222-229, 1963. Google Scholar
  17. Balázs Keszegh. Coloring half-planes and bottomless rectangles. Computational Geometry, 45(9):495-507, 2012. Google Scholar
  18. Balázs Keszegh, Nathan Lemons, and Dömötör Pálvölgyi. Online and quasi-online colorings of wedges and intervals. Order, 33(3):389-409, 2016. Google Scholar
  19. Balázs Keszegh and Dömötör Pálvölgyi. Octants are cover-decomposable. Discrete &Computational Geometry, 47(3):598-609, 2012. Google Scholar
  20. Balázs Keszegh and Dömötör Pálvölgyi. Convex polygons are self-coverable. Discrete &Computational Geometry, 51(4):885-895, 2014. Google Scholar
  21. Balázs Keszegh and Dömötör Pálvölgyi. Octants are cover-decomposable into many coverings. Computational Geometry, 47(5):585-588, 2014. Google Scholar
  22. Balázs Keszegh and Dömötör Pálvölgyi. An abstract approach to polychromatic coloring: Shallow hitting sets in aba-free hypergraphs and pseudohalfplanes. In Ernst W. Mayr, editor, Graph-Theoretic Concepts in Computer Science - 41st International Workshop, WG 2015, Garching, Germany, June 17-19, 2015, Revised Papers, volume 9224 of Lecture Notes in Computer Science, pages 266-280. Springer, 2015. Google Scholar
  23. Balázs Keszegh and Dömötör Pálvölgyi. More on decomposing coverings by octants. J. of Computational Geometry, 6(1):300-315, 2015. Google Scholar
  24. Rolf Klein. Concrete and abstract Voronoi diagrams, volume 400. Springer Science &Business Media, 1989. Google Scholar
  25. Jon M. Kleinberg, Rajeev Motwani, Prabhakar Raghavan, and Suresh Venkatasubramanian. Storage management for evolving databases. In 38th Annual Symposium on Foundations of Computer Science, FOCS'97, Miami Beach, Florida, USA, October 19-22, 1997, pages 353-362. IEEE Computer Society, 1997. Google Scholar
  26. István Kovács. Indecomposable coverings with homothetic polygons. Discrete &Computational Geometry, 53(4):817-824, 2015. Google Scholar
  27. L. Ma. Bisectors and Voronoi Diagrams for Convex Distance Functions. PhD thesis, FernUniversität Hagen, Germany, 2000. Google Scholar
  28. János Pach. Decomposition of multiple packing and covering. In 2. Kolloquium über Diskrete Geometrie, pages 169-178. Institut für Mathematik der Universität Salzburg, 1980. Google Scholar
  29. János Pach. Covering the plane with convex polygons. Discrete & Computational Geometry, 1:73-81, 1986. URL: http://dx.doi.org/10.1007/BF02187684.
  30. János Pach and Dömötör Pálvölgyi. Unsplittable coverings in the plane. Advances in Mathematics, 302:433-457, 2016. Google Scholar
  31. János Pach, Dömötör Pálvölgyi, and Géza Tóth. Survey on decomposition of multiple coverings. In Imre Bárány, Károly J. Böröczky, Gábor Fejes Tóth, and János Pach, editors, Geometry - Intuitive, Discrete, and Convex, volume 24 of Bolyai Society Mathematical Studies, pages 219-257. Springer Berlin Heidelberg, 2013. Google Scholar
  32. János Pach and Gábor Tardos. Coloring axis-parallel rectangles. J. of Combinatorial Theory, Series A, 117(6):776-782, 2010. URL: http://dx.doi.org/10.1016/j.jcta.2009.04.007.
  33. János Pach, Gábor Tardos, and Géza Tóth. Indecomposable coverings. Canadian mathematical bulletin, 52(3):451-463, 2009. Google Scholar
  34. János Pach and Géza Tóth. Decomposition of multiple coverings into many parts. In Proceedings of the twenty-third annual symposium on Computational geometry, pages 133-137. ACM, 2007. Google Scholar
  35. Dömötör Pálvölgyi. Decomposition of geometric set systems and graphs, phd thesis. arXiv preprint arXiv:1009.4641, 2010. Google Scholar
  36. Dömötör Pálvölgyi. Indecomposable coverings with concave polygons. Discrete & Computational Geometry, 44(3):577-588, 2010. URL: http://dx.doi.org/10.1007/s00454-009-9194-y.
  37. Dömötör Pálvölgyi and Géza Tóth. Convex polygons are cover-decomposable. Discrete &Computational Geometry, 43(3):483-496, 2010. Google Scholar
  38. K. S. Poh. On the linear vertex-arboricity of a planar graph. J. of Graph Theory, 14(1):73-75, 1990. Google Scholar
  39. Shakhar Smorodinsky. On the chromatic number of geometric hypergraphs. SIAM J. on Discrete Math., 21(3):676-687, 2007. Google Scholar
  40. Shakhar Smorodinsky and Yelena Yuditsky. Polychromatic coloring for half-planes. J. of Combinatorial Theory, Series A, 119(1):146-154, 2012. Google Scholar
  41. Gábor Tardos and Géza Tóth. Multiple coverings of the plane with triangles. Discrete & Computational Geometry, 38(2):443-450, 2007. URL: http://dx.doi.org/10.1007/s00454-007-1345-4.
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact