The Number of Crossings in Multigraphs with No Empty Lens

  • Michael Kaufmann
  • János Pach
  • Géza Tóth
  • Torsten UeckerdtEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11282)


Let G be a multigraph with n vertices and \(e>4n\) edges, drawn in the plane such that any two parallel edges form a simple closed curve with at least one vertex in its interior and at least one vertex in its exterior. Pach and Tóth [5] extended the Crossing Lemma of Ajtai et al. [1] and Leighton [3] by showing that if no two adjacent edges cross and every pair of nonadjacent edges cross at most once, then the number of edge crossings in G is at least \(\alpha e^3/n^2\), for a suitable constant \(\alpha >0\). The situation turns out to be quite different if nonparallel edges are allowed to cross any number of times. It is proved that in this case the number of crossings in G is at least \(\alpha e^{2.5}/n^{1.5}\). The order of magnitude of this bound cannot be improved.



This project initiated at the Dagstuhl seminar 16452 “Beyond-Planar Graphs: Algorithmics and Combinatorics,” November 2016. We would like to thank all participants, especially Stefan Felsner, Vincenzo Roselli, and Pavel Valtr, for fruitful discussions.


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Copyright information

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Authors and Affiliations

  • Michael Kaufmann
    • 1
  • János Pach
    • 2
    • 3
  • Géza Tóth
    • 3
  • Torsten Ueckerdt
    • 4
    Email author
  1. 1.Wilhelm-Schickard-Institut für InformatikUniversität TübingenTübingenGermany
  2. 2.EPFLLausanneSwitzerland
  3. 3.Rényi InstituteBudapestHungary
  4. 4.Institute of Theoretical InformaticsKarlsruhe Institute of Technology (KIT)KarlsruheGermany

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