Coloring Intersection Hypergraphs of Pseudo-Disks

Author Balázs Keszegh



PDF
Thumbnail PDF

File

LIPIcs.SoCG.2018.52.pdf
  • Filesize: 0.56 MB
  • 15 pages

Document Identifiers

Author Details

Balázs Keszegh

Cite AsGet BibTex

Balázs Keszegh. Coloring Intersection Hypergraphs of Pseudo-Disks. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 52:1-52:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.SoCG.2018.52

Abstract

We prove that the intersection hypergraph of a family of n pseudo-disks with respect to another family of pseudo-disks admits a proper coloring with 4 colors and a conflict-free coloring with O(log n) colors. Along the way we prove that the respective Delaunay-graph is planar. We also prove that the intersection hypergraph of a family of n regions with linear union complexity with respect to a family of pseudo-disks admits a proper coloring with constantly many colors and a conflict-free coloring with O(log n) colors. Our results serve as a common generalization and strengthening of many earlier results, including ones about proper and conflict-free coloring points with respect to pseudo-disks, coloring regions of linear union complexity with respect to points and coloring disks with respect to disks.
Keywords
  • combinatorial geometry
  • conflict-free coloring
  • geometric hypergraph coloring

Metrics

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

References

  1. Eyal Ackerman, Balázs Keszegh, and Mate Vizer. Coloring points with respect to squares. Discrete & Computational Geometry, jun 2017. Google Scholar
  2. B. Aronov, A. Donakonda, E. Ezra, and R. Pinchasi. On Pseudo-disk Hypergraphs. ArXiv e-prints, 2018. URL: http://arxiv.org/abs/1802.08799.
  3. Sarit Buzaglo, Rom Pinchasi, and Günter Rote. Topological hypergraphs. In Thirty Essays on Geometric Graph Theory, pages 71-81. Springer New York, oct 2012. Google Scholar
  4. Jean Cardinal and Matias Korman. Coloring planar homothets and three-dimensional hypergraphs. Computational geometry, 46(9):1027-1035, 2013. Google Scholar
  5. Timothy M. Chan and Sariel Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. Discrete & Computational Geometry, 48(2):373-392, 2012. URL: http://dx.doi.org/10.1007/s00454-012-9417-5.
  6. Chaim Chojnacki. Über wesentlich unplättbare kurven im dreidimensionalen raume. Fundamenta Mathematicae, 23(1):135-142, 1934. Google Scholar
  7. Guy Even, Zvi Lotker, Dana Ron, and Shakhar Smorodinsky. Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks. SIAM Journal on Computing, 33(1):94-136, jan 2003. Google Scholar
  8. S. P. Fekete and P. Keldenich. Conflict-Free Coloring of Intersection Graphs. ArXiv e-prints, 2017. URL: http://arxiv.org/abs/1709.03876.
  9. Andreas F. Holmsen, Hossein Nassajian Mojarrad, János Pach, and Gábor Tardos. Two extensions of the Erdős-Szekeres problem. ArXiv e-prints, 2017. URL: http://arxiv.org/abs/1710.11415.
  10. Klara Kedem, Ron Livne, János Pach, and Micha Sharir. On the union of jordan regions and collision-free translational motion amidst polygonal obstacles. Discrete & Computational Geometry, 1(1):59-71, Mar 1986. Google Scholar
  11. C. Keller and S. Smorodinsky. Conflict-Free Coloring of Intersection Graphs of Geometric Objects. ArXiv e-prints, 2017. URL: http://arxiv.org/abs/1704.02018.
  12. Balázs Keszegh. Weak conflict-free colorings of point sets and simple regions. In Prosenjit Bose, editor, Proceedings of the 19th Annual Canadian Conference on Computational Geometry, CCCG 2007, August 20-22, 2007, Carleton University, Ottawa, Canada, pages 97-100. Carleton University, Ottawa, Canada, 2007. URL: http://cccg.ca/proceedings/2007/04b2.pdf.
  13. Balázs Keszegh. Coloring half-planes and bottomless rectangles. Computational geometry, 45(9):495-507, 2012. URL: http://dx.doi.org/10.1016/j.comgeo.2011.09.004.
  14. Balázs Keszegh and Dömötör Pálvölgyi. Octants are cover-decomposable. Discrete & Computational Geometry, 47(3):598-609, 2012. Google Scholar
  15. 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. URL: http://dx.doi.org/10.1007/978-3-662-53174-7_19.
  16. Balázs Keszegh and Dömötör Pálvölgyi. More on decomposing coverings by octants. Journal of Computational Geometry, 6(1):300-315, 2015. Google Scholar
  17. Balázs Keszegh and Dömötör Pálvölgyi. Proper coloring of geometric hypergraphs. In Symposium on Computational Geometry, volume 77 of LIPIcs, pages 47:1-47:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. Google Scholar
  18. István Kovács. Indecomposable coverings with homothetic polygons. Discrete &Computational Geometry, 53(4):817-824, 2015. Google Scholar
  19. János Pach and Dömötör Pálvölgyi. Unsplittable coverings in the plane. 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 281-296. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-53174-7_20.
  20. János Pach, Dömötör Pálvölgyi, and Géza Tóth. Survey on decomposition of multiple coverings. In Geometry—Intuitive, Discrete, and Convex, pages 219-257. Springer, 2013. Google Scholar
  21. János Pach and Micha Sharir. On the boundary of the union of planar convex sets. Discrete &Computational Geometry, 21(3):321-328, 1999. Google Scholar
  22. János Pach, Gábor Tardos, and Géza Tóth. Indecomposable coverings. In Discrete Geometry, Combinatorics and Graph Theory, pages 135-148. Springer, 2007. Google Scholar
  23. Rom Pinchasi. A finite family of pseudodiscs must include a quotedblleftsmallquotedblright pseudodisc. SIAM Journal on Discrete Mathematics, 28(4):1930-1934, jan 2014. Google Scholar
  24. Fulek Radoslav, Michael J Pelsmajer, Marcus Schaefer, and Daniel Štefankovič. Adjacent crossings do matter. Journal of Graph Algorithms and Applications, 16(3):759-782, 2012. Google Scholar
  25. Shakhar Smorodinsky. On the chromatic number of geometric hypergraphs. SIAM Journal on Discrete Mathematics, 21(3):676-687, 2007. Google Scholar
  26. Shakhar Smorodinsky. Conflict-free coloring and its applications. In Geometry—Intuitive, Discrete, and Convex, pages 331-389. Springer, 2013. Google Scholar
  27. Jack Snoeyink and John Hershberger. Sweeping arrangements of curves. In Proceedings of the fifth annual symposium on Computational geometry, pages 354-363. ACM, 1989. Google Scholar
  28. Géza Tóth. personal communication. Google Scholar
  29. William T Tutte. Toward a theory of crossing numbers. Journal of Combinatorial Theory, 8(1):45-53, 1970. Google Scholar