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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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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

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