Document Open Access Logo

Improving the Crossing Lemma by Characterizing Dense 2-Planar and 3-Planar Graphs

Authors Aaron Büngener, Michael Kaufmann

PDF
Thumbnail PDF

File

LIPIcs.GD.2024.29.pdf
  • Filesize: 1.1 MB
  • 22 pages

Document Identifiers

Author Details

Aaron Büngener
  • Universität Tübingen, Germany
Michael Kaufmann
  • Universität Tübingen, Germany

Cite As Get BibTex

Aaron Büngener and Michael Kaufmann. Improving the Crossing Lemma by Characterizing Dense 2-Planar and 3-Planar Graphs. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 29:1-29:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.GD.2024.29

Abstract

The classical Crossing Lemma by Ajtai et al. and Leighton from 1982 gave an important lower bound of cm³/n² for the number of crossings in any drawing of a given graph of n vertices and m edges. The original value was c = 1/100, which then has gradually been improved. Here, the bounds for the density of k-planar graphs played a central role. Our new insight is that for k = 2,3 the k-planar graphs have substantially fewer edges if specific local configurations that occur in drawings of k-planar graphs of maximum density are forbidden. Therefore, we are able to derive better bounds for the crossing number cr(G) of a given graph G. In particular, we achieve a bound of cr(G) ≥ 73/18m-305/18(n-2) for the range of 5n < m ≤ 6n, while our second bound cr(G) ≥ 5m - 407/18(n-2) is even stronger for larger m > 6n.
For m > 6.79n, we finally apply the standard probabilistic proof from the BOOK and obtain an improved constant of c > 1/27.61 in the Crossing Lemma. Note that the previous constant was 1/29. Although this improvement is not too impressive, we consider our technique as an important new tool, which might be helpful in various other applications.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Combinatorics
Keywords
  • Crossing Lemma
  • k-planar graphs
  • discharging method

Metrics

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

References

  1. Eyal Ackerman. On topological graphs with at most four crossings per edge. Comput. Geom., 85, 2019. URL: https://doi.org/10.1016/J.COMGEO.2019.101574.
  2. Eyal Ackerman and Gábor Tardos. On the maximum number of edges in quasi-planar graphs. J. Comb. Theory, Ser. A, 114(3):563-571, 2007. URL: https://doi.org/10.1016/J.JCTA.2006.08.002.
  3. Martin Aigner and Günter M. Ziegler. Proofs from THE BOOK (3. ed.). Springer, 2004. Google Scholar
  4. Miklós Ajtai, Vašek Chvátal, Monroe M Newborn, and Endre Szemerédi. Crossing-free subgraphs. In North-Holland Mathematics Studies, volume 60, pages 9-12. Elsevier, 1982. Google Scholar
  5. Patrizio Angelini, Michael A. Bekos, Michael Kaufmann, Maximilian Pfister, and Torsten Ueckerdt. Beyond-planarity: Density results for bipartite graphs. CoRR, abs/1712.09855, 2017. URL: https://arxiv.org/abs/1712.09855.
  6. Sang Won Bae, Jean-François Baffier, Jinhee Chun, Peter Eades, Kord Eickmeyer, Luca Grilli, Seok-Hee Hong, Matias Korman, Fabrizio Montecchiani, Ignaz Rutter, and Csaba D. Tóth. Gap-planar graphs. Theor. Comput. Sci., 745:36-52, 2018. URL: https://doi.org/10.1016/J.TCS.2018.05.029.
  7. Michael A. Bekos, Michael Kaufmann, and Chrysanthi N. Raftopoulou. On optimal 2- and 3-planar graphs. In Boris Aronov and Matthew J. Katz, editors, 33rd International Symposium on Computational Geometry, SoCG 2017, July 4-7, 2017, Brisbane, Australia, volume 77 of LIPIcs, pages 16:1-16:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPICS.SOCG.2017.16.
  8. Carla Binucci, Aaron Büngener, Giuseppe Di Battista, Walter Didimo, Vida Dujmovic, Seok-Hee Hong, Michael Kaufmann, Giuseppe Liotta, Pat Morin, and Alessandra Tappini. Min-k-planar drawings of graphs. In Michael A. Bekos and Markus Chimani, editors, Graph Drawing and Network Visualization - 31st International Symposium, GD 2023, Palermo, Italy, September 20-22, 2023, Revised Selected Papers, Part I, volume 14465 of Lecture Notes in Computer Science, pages 39-52. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-49272-3_3.
  9. Michael Kaufmann, János Pach, Géza Tóth, and Torsten Ueckerdt. The number of crossings in multigraphs with no empty lens. J. Graph Algorithms Appl., 25(1):383-396, 2021. URL: https://doi.org/10.7155/JGAA.00563.
  10. Frank Thomson Leighton. Complexity issues in VLSI: optimal layouts for the shuffle-exchange graph and other networks. MIT press, 1983. Google Scholar
  11. János Pach, Rados Radoicic, Gábor Tardos, and Géza Tóth. Improving the crossing lemma by finding more crossings in sparse graphs. Discret. Comput. Geom., 36(4):527-552, 2006. URL: https://doi.org/10.1007/S00454-006-1264-9.
  12. János Pach, Joel Spencer, and Géza Tóth. New bounds on crossing numbers. Discret. Comput. Geom., 24(4):623-644, 2000. URL: https://doi.org/10.1007/S004540010011.
  13. János Pach and Géza Tóth. Graphs drawn with few crossings per edge. Comb., 17(3):427-439, 1997. URL: https://doi.org/10.1007/BF01215922.
  14. János Pach and Géza Tóth. A crossing lemma for multigraphs. Discret. Comput. Geom., 63(4):918-933, 2020. URL: https://doi.org/10.1007/S00454-018-00052-Z.
  15. Radoš Radoičić and Géza Tóth. The discharging method in combinatorial geometry and the pach-sharir conjecture. In Surveys on Discrete and Computational Geometry: Twenty Years Later: AMS-IMS-SIAM Joint Summer Research Conference, June 18-22, 2006, Snowbird, Utah, volume 453, page 319. American Mathematical Soc., 2008. Google Scholar
  16. Marcus Schaefer. The graph crossing number and its variants: A survey. The electronic journal of combinatorics, pages DS21-Apr, 2012. Google Scholar
  17. László A. Székely. Crossing numbers and hard erdös problems in discrete geometry. Comb. Probab. Comput., 6(3):353-358, 1997. URL: http://journals.cambridge.org/action/displayAbstract?aid=46513.
  18. R Von Bodendiek, Heinz Schumacher, and Klaus Wagner. Bemerkungen zu einem sechsfarbenproblem von g. ringel. In Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, volume 53, pages 41-52. Springer, 1983. Google Scholar
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 About DROPS Imprint Privacy Contact