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A First View on the Density of 5-Planar Graphs

Authors Aaron Büngener , Jakob Franz, Michael Kaufmann , Maximilian Pfister

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ACM Subject Classification
  • Mathematics of computing → Graph theory
Keywords
  • k-planar graphs
  • 5-planarity
  • edge density
  • Crossing Lemma
  • discharging

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Abstract

A key concept for many graph layout algorithms is planarity, a graph property that allows to draw vertices and edges crossing-free in the plane. Important is the generalization to k-planar graphs, which can be drawn in the plane with at most k > 0 crossings per edge. One of the basic graph properties that have been explored for those graph classes is the maximum edge density, i.e., the maximum number of edges a k-planar graph on n vertices may have. While there are numerous results for the classes of 1- and 2-planar graphs, there are few results for increasing k = 3 or 4 due to the complex graph structures. We make a first step towards even larger k > 4 exploring the class of 5-planar graphs. While our main tool is still a discharging technique, a better understanding of the structure of the denser parts leads to corresponding density bounds in a much simpler way.
We first apply a simplified version of our technique to outer 5-planar graphs and surprisingly observe that the structure of maximally dense (general) 5-planar graphs differs from the known uniform structure of maximally dense k-planar graphs for smaller k ∈ {1,2,3,4}. As the central result of this paper, we then show that graphs that admit a simple 5-planar drawing have at most 7(n-2) edges, drastically improving the previous best bound of ≈8.3n. This even implies a small improvement of the leading constant in the Crossing Lemma cr(G) ≥ c m³/n² from c = 1/(27.48) to c = 1/(27.3). To demonstrate the potential of our new technique, we also apply it to 4-planar and 6-planar graphs.

Cite As Get BibTex

Aaron Büngener, Jakob Franz, Michael Kaufmann, and Maximilian Pfister. A First View on the Density of 5-Planar Graphs. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.GD.2026.33

Author Details

Aaron Büngener
  • University of Tübingen, Germany
Jakob Franz
  • University of Tübingen, Germany
Michael Kaufmann
  • University of Tübingen, Germany
Maximilian Pfister
  • University of Tübingen, Germany

Funding

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

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