A Crossing Lemma for Multigraphs

Authors János Pach, Géza Tóth



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János Pach
Géza Tóth

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János Pach and Géza Tóth. A Crossing Lemma for Multigraphs. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 65:1-65:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.SoCG.2018.65

Abstract

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.
Keywords
  • crossing number
  • Crossing Lemma
  • multigraph
  • separator theorem

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