Document Open Access Logo

On the Maximum Number of Tangencies Among 1-Intersecting Curves

Authors Eyal Ackerman , Balázs Keszegh

PDF

File

Thumbnail PDF
PDF
LIPIcs.SoCG.2026.2.pdf
  • Filesize: 0.83 MB
  • 15 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Extremal graph theory
  • Mathematics of computing → Graphs and surfaces
Keywords
  • tangency graph
  • forbidden subgraph
  • extremal graph

Metrics

Abstract

According to a conjecture of Pach, there are O(n) tangent pairs among any family of n Jordan arcs in which every pair of arcs has precisely one common point and no three arcs share a common point. This conjecture was proved for two special cases, however, for the general case the currently best upper bound is only O(n^{7/4}). This is also the best known bound on the number of tangencies in the relaxed case where every pair of arcs has at most one common point. We improve the bounds for the latter and former cases to O(n^{5/3}) and O(n^{3/2}), respectively. We also consider a few other variants of these questions, for example, we show that if the arcs are x-monotone, each pair intersects at most once and their left endpoints lie on a common vertical line, then the maximum number of tangencies is Θ(n^{4/3}). Without this last condition the number of tangencies is O(n^{4/3}(log n)^{1/3}), improving a previous bound of Pach and Sharir. Along the way we prove a graph-theoretic theorem which extends a result of Erdős and Simonovits and may be of independent interest.

Cite As Get BibTex

Eyal Ackerman and Balázs Keszegh. On the Maximum Number of Tangencies Among 1-Intersecting Curves. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 2:1-2:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.2

Author Details

Eyal Ackerman
  • Department of Mathematics, Physics and Computer Science, University of Haifa at Oranim, Israel
Balázs Keszegh
  • HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
  • ELTE Eötvös Loránd University, Budapest, Hungary

Funding

  • Keszegh, Balázs: Supported by the ERC Advanced Grant "ERMiD", no. 101054936 and by the EXCELLENCE-24 project no. 151504 Combinatorics and Geometry of the NRDI Fund.

Acknowledgements

We are grateful to Rom Pinchasi for many helpful discussions on the problems studied in this paper. In particular he has suggested most of the ingredients of the proofs of Theorems 7 and 5 and later found out that similar results were already proved by Pach and Sharir [Pach and Sharir, 1991]. We thank Balázs Patkós for his comments about Theorem 9. For the proof of Theorem 10 we used some back and forth interaction with Google’s Large Language Model "Gemini" (https://gemini.google.com/).

References

  1. Eyal Ackerman, Gábor Damásdi, Balázs Keszegh, Rom Pinchasi, and Rebeka Raffay. The Maximum Number of Digons Formed by Pairwise Intersecting Pseudocircles. In Oswin Aichholzer and Haitao Wang, editors, 41st International Symposium on Computational Geometry (SoCG 2025), volume 332 of Leibniz International Proceedings in Informatics (LIPIcs), pages 2:1-2:14, Dagstuhl, Germany, 2025. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.SoCG.2025.2.
  2. Eyal Ackerman and Balázs Keszegh. On the maximum number of tangencies among 1-intersecting curves, 2026. URL: https://arxiv.org/abs/2603.11885.
  3. Eyal Ackerman, Balázs Keszegh, and Dömötör Pálvölgyi. On tangencies among planar curves with an application to coloring L-shapes. European Journal of Combinatorics, 121:103837, 2024. URL: https://doi.org/10.1016/j.ejc.2023.103837.
  4. Eyal Ackerman and Balázs Keszegh. On the number of tangencies among 1-intersecting x-monotone curves. European Journal of Combinatorics, 118:103929, 2024. URL: https://doi.org/10.1016/j.ejc.2024.103929.
  5. Pankaj K. Agarwal, Eran Nevo, János Pach, Rom Pinchasi, Micha Sharir, and Shakhar Smorodinsky. Lenses in arrangements of pseudo-circles and their applications. J. ACM, 51(2):139-186, 2004. URL: https://doi.org/10.1145/972639.972641.
  6. Itay Ben-Dan and Rom Pinchasi. personal communication. Google Scholar
  7. Jon Bentley and Thomas Ottmann. Algorithms for reporting and counting geometric intersections. IEEE Transactions on Computers, C-28(9):643-647, 1979. URL: https://doi.org/10.1109/TC.1979.1675432.
  8. Peter Brass, William O. J. Moser, and János Pach. Research problems in discrete geometry. Springer, 2005. Google Scholar
  9. Jordan S. Ellenberg, Jozsef Solymosi, and Joshua Zahl. New bounds on curve tangencies and orthogonalities. Discrete Analysis, November 4 2016. URL: https://doi.org/10.19086/da.990.
  10. P. Erdős and M. Simonovits. Some extremal problems in graph theory. In P. Erdős, A. Rényi, and V. T. Sós, editors, Combinatorial Theory and its Applications, I, volume 4 of Colloquia Mathematica Societatis János Bolyai, pages 377-390. North-Holland, Amsterdam, 1970. Proc. Colloq., Balatonfüred (Hungary), 1969. URL: https://users.renyi.hu/~miki/1970-22ErdSimCube.pdf.
  11. Pál Erdős and Branko Grünbaum. Osculation vertices in arrangements of curves. Geometriae Dedicata, 1(4):322-333, 1973. URL: https://doi.org/10.1007/BF00147765.
  12. István Fáry. On straight line-representations of planar graphs. Acta Scientiarum Mathematicarum, 12(182-192):229, 1948. Google Scholar
  13. Branko Grünbaum. Arrangements and spreads. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 10. American Mathematical Society, Providence, R.I., 1972. Google Scholar
  14. Péter Györgyi, Bálint Hujter, and Sándor Kisfaludi-Bak. On the number of touching pairs in a set of planar curves. Computational Geometry, 67:29-37, 2018. URL: https://doi.org/10.1016/j.comgeo.2017.10.004.
  15. Sariel Har-Peled. Constructing planar cuttings in theory and practice. SIAM Journal on Computing, 29(6):2016-2039, 2000. URL: https://doi.org/10.1137/S0097539799350232.
  16. Barnabás Janzer, Oliver Janzer, Abhishek Methuku, and Gábor Tardos. Tight bounds for intersection-reverse sequences, edge-ordered graphs and applications. Journal of the London Mathematical Society, 112(4), October 2025. URL: https://doi.org/10.1112/jlms.70324.
  17. Balázs Keszegh and Dömötör Pálvölgyi. The number of tangencies between two families of curves. Combinatorica, 43(5):939-952, October 2023. URL: https://doi.org/10.1007/s00493-023-00041-8.
  18. T. Kővári, V. Sós, and P. Turán. On a problem of K. Zarankiewicz. Colloquium Mathematicae, 3(1):50-57, 1954. URL: http://eudml.org/doc/210011.
  19. Adam Marcus and Gábor Tardos. Intersection reverse sequences and geometric applications. J. Comb. Theory, Ser. A, 113(4):675-691, 2006. URL: https://doi.org/10.1016/j.jcta.2005.07.002.
  20. János Pach. personal communication. Google Scholar
  21. János Pach and Pankaj K. Agarwal. Combinatorial Geometry, chapter 11, pages 177-178. John Wiley and Sons Ltd, 1995. URL: https://doi.org/10.1002/9781118033203.ch11.
  22. János Pach, Natan Rubin, and Gábor Tardos. A crossing lemma for Jordan curves. Advances in Mathematics, 331:908-940, 2018. URL: https://doi.org/10.1016/j.aim.2018.03.015.
  23. János Pach and Micha Sharir. On vertical visibility in arrangements of segments and the queue size in the Bentley-Ottmann line sweeping algorithm. SIAM Journal on Computing, 20(3):460-470, 1991. URL: https://doi.org/10.1137/0220029.
  24. János Pach and Micha Sharir. Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures, volume 152 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2009. URL: https://doi.org/10.1090/surv/152.
  25. János Pach, Andrew Suk, and Miroslav Treml. Tangencies between families of disjoint regions in the plane. Comput. Geom., 45(3):131-138, 2012. URL: https://doi.org/10.1016/j.comgeo.2011.10.002.
  26. Rom Pinchasi and Radoš Radoičić. On the number of edges in a topological graph with no self-intersecting cycle of length 4. In János Pach, editor, Towards a Theory of Geometric Graphs, volume 342 of Contemporary Mathematics, pages 233-234. American Mathematical Society, 2004. Google Scholar
  27. Micha Sharir and Pankaj K. Agarwal. Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, New York, 1995. 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