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

Authors Aaron Büngener, Michael Kaufmann

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LIPIcs.GD.2024.29.pdf
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Subject Classification

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

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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.

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

Author Details

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

References

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