A Crossing Lemma for Multigraphs

  • János Pach
  • Géza TóthEmail author


Let G be a drawing of a graph with n vertices and \(e>4n\) edges, in which no two adjacent edges cross and any pair of independent edges cross at most once. According to the celebrated Crossing Lemma of Ajtai, Chvátal, Newborn, Szemerédi and Leighton, the number of crossings in G is at least \(c\,{e^3\over n^2}\), for a suitable constant \(c>0\). In a seminal paper, Székely generalized this result to multigraphs, establishing the lower bound \(c\,{e^3\over mn^2}\), where m denotes the maximum multiplicity of an edge in G. We get rid of the dependence on m by showing that, as in the original Crossing Lemma, the number of crossings is at least \(c'{e^3\over n^2}\) for some \(c'>0\), provided that the “lens” enclosed by every pair of parallel edges in G contains at least one vertex. This settles a conjecture of Bekos, Kaufmann, and Raftopoulou.


Crossing number Crossing Lemma Separator theorem Bisection width 

Mathematics Subject Classification




We are very grateful to Stefan Felsner, Michael Kaufmann, Vincenzo Roselli, Torsten Ueckerdt, and Pavel Valtr for their many valuable comments, suggestions, and for many interesting discussions during the Dagstuhl Seminar “Beyond-Planar Graphs: Algorithmics and Combinatorics”, November 6–11, 2016. We also want to thank Felix Schröder for his comments and for carefully reading our manuscript. János Pach: Supported by Swiss National Science Foundation Grants 200021-165977 and 200020-162884 and Schloss Dagstuhl – Leibniz Center for Informatics. Géza Tóth: Supported by National Research, Development and Innovation Office, NKFIH, K-111827 and Schloss Dagstuhl – Leibniz Center for Informatics. This work is connected to the scientific program of the “Development of quality-oriented and harmonized R+D+I strategy and functional model at BME” project, supported by the New Hungary Development Plan (Project ID: TÁMOP-4.2.1/B-09/1/KMR-2010-0002).


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Authors and Affiliations

  1. 1.Ecole Polytechnique Fédérale de Lausanne and Rényi InstituteHungarian Academy of SciencesBudapestHungary
  2. 2.Rényi InstituteHungarian Academy of SciencesBudapestHungary
  3. 3.Department of Mathematical AnalysisBudapest University of Technology and EconomicsBudapestHungary

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