On topological graphs with at most four crossings per edge

Research output: Contribution to journalArticlepeer-review


We show that if a graph G with n≥3 vertices can be drawn in the plane such that each of its edges is involved in at most four crossings, then G has at most 6n−12 edges. This settles a conjecture of Pach, Radoičić, Tardos, and Tóth, and yields a better bound for the famous Crossing Lemma: The crossing number, cr(G), of a (not too sparse) graph G with n vertices and m edges is at least [Formula presented], where c>1/29. This bound is known to be tight, apart from the constant c for which the previous best lower bound was 1/31.1.

Original languageEnglish
Article number101574
JournalComputational Geometry: Theory and Applications
StatePublished - Dec 2019

Bibliographical note

Publisher Copyright:
© 2019 Elsevier B.V.


  • Crossing lemma
  • Topological graphs
  • k-Planar graphs

ASJC Scopus subject areas

  • Computer Science Applications
  • Geometry and Topology
  • Control and Optimization
  • Computational Theory and Mathematics
  • Computational Mathematics


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