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On k-Planar Graphs Without Short Cycles

Authors Michael A. Bekos , Prosenjit Bose , Aaron Büngener, Vida Dujmović , Michael Hoffmann , Michael Kaufmann , Pat Morin , Saeed Odak, Alexandra Weinberger

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LIPIcs.GD.2024.27.pdf
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ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Graph theory
  • Human-centered computing → Graph drawings
Keywords
  • Beyond-planar Graphs
  • k-planar Graphs
  • Local Crossing Number
  • Crossing Number
  • Discharging Method
  • Crossing Lemma

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Abstract

We study the impact of forbidding short cycles to the edge density of k-planar graphs; a k-planar graph is one that can be drawn in the plane with at most k crossings per edge. Specifically, we consider three settings, according to which the forbidden substructures are 3-cycles, 4-cycles or both of them (i.e., girth ≥ 5). For all three settings and all k ∈ {1,2,3}, we present lower and upper bounds on the maximum number of edges in any k-planar graph on n vertices. Our bounds are of the form c n, for some explicit constant c that depends on k and on the setting. For general k ≥ 4 our bounds are of the form c√kn, for some explicit constant c. These results are obtained by leveraging different techniques, such as the discharging method, the recently introduced density formula for non-planar graphs, and new upper bounds for the crossing number of 2- and 3-planar graphs in combination with corresponding lower bounds based on the Crossing Lemma.

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Michael A. Bekos, Prosenjit Bose, Aaron Büngener, Vida Dujmović, Michael Hoffmann, Michael Kaufmann, Pat Morin, Saeed Odak, and Alexandra Weinberger. On k-Planar Graphs Without Short Cycles. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.GD.2024.27

Author Details

Michael A. Bekos
  • Department of Mathematics, University of Ioannina, Greece
Prosenjit Bose
  • School of Computer Science, Carleton University, Ottawa, Canada
Aaron Büngener
  • Department of Computer Science, University of Tübingen, Germany
Vida Dujmović
  • School of Electrical Engineering and Computer Science, University of Ottawa, Canada
Michael Hoffmann
  • Department of Computer Science, ETH Zürich, Switzerland
Michael Kaufmann
  • Department of Computer Science, University of Tübingen, Germany
Pat Morin
  • School of Computer Science, Carleton University, Ottawa, Canada
Saeed Odak
  • School of Electrical Engineering and Computer Science, University of Ottawa, Canada
Alexandra Weinberger
  • Faculty of Mathematics and Computer Science, FernUniversität in Hagen, Germany

Funding

  • Weinberger, Alexandra: Part of this work was done while A. W. was at the University of Technology in Graz and supported by the Austrian Science Fund (FWF): W1230.

Acknowledgements

This work was started at the Summer Workshop on Graph Drawing (SWGD 2023) in Caldana, Italy.

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. Springer, Berlin, 4th edition, 2010. URL: https://doi.org/10.1007/978-3-642-00856-6.
  4. Miklós Ajtai, Václav Chvátal, Monroe M. Newborn, and Endre Szemerédi. Crossing-free subgraphs. Ann. Discrete Math., 12:9-12, 1982. URL: https://doi.org/10.1016/S0304-0208(08)73484-4.
  5. Patrizio Angelini, Michael A. Bekos, Michael Kaufmann, Maximilian Pfister, and Torsten Ueckerdt. Beyond-planarity: Turán-type results for non-planar bipartite graphs. In Proc. 29th Annu. Internat. Sympos. Algorithms Comput., volume 123 of LIPIcs, pages 28:1-28:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPICS.ISAAC.2018.28.
  6. Michael A. Bekos, Prosenjit Bose, Aaron Büngener, Vida Dujmović, Michael Hoffmann, Michael Kaufmann, Pat Morin, Saeed Odak, and Alexandra Weinberger. On k-planar graphs without short cycles. CoRR, abs/2408.16085, 2024. URL: https://arxiv.org/abs/2408.16085.
  7. Michael A. Bekos, Michael Kaufmann, and Chrysanthi N. Raftopoulou. On optimal 2- and 3-planar graphs. In Proc. 33rd Internat. Sympos. Comput. Geom., 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. Rainer Bodendiek, Heinz Schumacher, and Klaus Wagner. Bemerkungen zu einem Sechsfarbenproblem von G. Ringel. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 53:41-52, 1983. URL: https://doi.org/10.1007/BF02941309.
  9. Franz J. Brandenburg. Recognizing optimal 1-planar graphs in linear time. Algorithmica, 80(1):1-28, 2018. URL: https://doi.org/10.1007/S00453-016-0226-8.
  10. Július Czap, Jakub Przybylo, and Erika Skrabul'áková. On an extremal problem in the class of bipartite 1-planar graphs. Discuss. Math. Graph Theory, 36(1):141-151, 2016. URL: https://doi.org/10.7151/DMGT.1845.
  11. Chris Dowden. Extremal C₄-free/C₅-free planar graphs. J. Graph Theory, 83(3):213-230, 2016. URL: https://doi.org/10.1002/JGT.21991.
  12. Jacob Fox, János Pach, and Csaba D. Tóth. A bipartite strengthening of the crossing lemma. In Proc. 15th Int. Sympos. Graph Drawing Network Visualization (GD 2007), volume 4875 of Lecture Notes in Computer Science, pages 13-24. Springer, 2007. URL: https://doi.org/10.1007/978-3-540-77537-9_4.
  13. Jacob Fox, János Pach, and Csaba D. Tóth. A bipartite strengthening of the crossing lemma. J. Comb. Theory B, 100(1):23-35, 2010. URL: https://doi.org/10.1016/J.JCTB.2009.03.005.
  14. Zoltán Füredi. New asymptotics for bipartite Turán numbers. J. Comb. Theory, Ser. A, 75(1):141-144, 1996. URL: https://doi.org/10.1006/JCTA.1996.0067.
  15. Seok-Hee Hong and Takeshi Tokuyama, editors. Beyond Planar Graphs. Springer, 2020. URL: https://doi.org/10.1007/978-981-15-6533-5.
  16. Michael Kaufmann, Boris Klemz, Kristin Knorr, Meghana M. Reddy, Felix Schröder, and Torsten Ueckerdt. The density formula: One lemma to bound them all. In Proc. 32nd Int. Sympos. Graph Drawing Network Visualization (GD 2024), volume 320 of LIPIcs, pages 5:1-5:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.GD.2024.5.
  17. János Pach, Rados Radoicic, Gábor Tardos, and Géza Tóth. Improving the crossing lemma by finding more crossings in sparse graphs: [extended abstract]. In Proc. 20th Annu. Sympos. Comput. Geom., pages 68-75. ACM, 2004. URL: https://doi.org/10.1145/997817.997831.
  18. János Pach, Radoš Radoičić, Gábor Tardos, and Géza Tóth. Improving the crossing lemma by finding more crossings in sparse graphs. Discrete Comput. Geom., 36(4):527-552, 2006. URL: https://doi.org/10.1007/s00454-006-1264-9.
  19. János Pach, Joel Spencer, and Géza Tóth. New bounds on crossing numbers. In Proc. 15th Annu. Sympos. Comput. Geom., pages 124-133. ACM, 1999. URL: https://doi.org/10.1145/304893.304943.
  20. 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.
  21. János Pach and Géza Tóth. Graphs drawn with few crossings per edge. In Stephen C. North, editor, Proc. 4th Sympos. Graph Drawing (GD 1996), volume 1190 of Lecture Notes in Computer Science, pages 345-354. Springer, 1996. URL: https://doi.org/10.1007/3-540-62495-3_59.
  22. János Pach and Géza Tóth. Graphs drawn with few crossings per edge. Combinatorica, 17(3):427-439, 1997. URL: https://doi.org/10.1007/BF01215922.
  23. Rephael Wenger. Extremal graphs with no C⁴’s, C⁶’s, or C^10’s. J. Comb. Theory, Ser. B, 52(1):113-116, 1991. URL: https://doi.org/10.1016/0095-8956(91)90097-4.
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