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On the Edge Density of Bipartite 3-Planar and Bipartite Gap-Planar Graphs

Authors Aaron Büngener, Maximilian Pfister

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Graph theory
Keywords
  • Edge Density
  • Beyond Planarity
  • bipartite Graphs
  • Discharging Method

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Abstract

We show that if a bipartite graph G with n ≥ 3 vertices can be drawn in the plane such that (i) each edge is involved in at most three crossings per edge or (ii) each crossing is assigned to one of the two involved edges and each edge is assigned at most one crossing, then G has at most 4n-8 edges. In both cases, this bound is tight up to an additive constant as witnessed by lower-bound constructions. The former result can be used to improve the leading constant for the crossing lemma for bipartite graphs which in turn improves various results such as the biplanar crossing number or the maximum number of edges a bipartite k-planar graph can have.

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Aaron Büngener and Maximilian Pfister. On the Edge Density of Bipartite 3-Planar and Bipartite Gap-Planar Graphs. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 28:1-28:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.GD.2024.28

Author Details

Aaron Büngener
  • Universität Tübingen, Germany
Maximilian Pfister
  • Universität Tübingen, Germany

Funding

  • Pfister, Maximilian: This research was supported by the DFG grant SCHL 2331/1-1.

Acknowledgements

We thank the anonymous reviewers for their valuable feedback.

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