A Bipartite Strengthening of the Crossing Lemma

  • Jacob Fox
  • János Pach
  • Csaba D. Tóth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4875)

Abstract

The celebrated Crossing Lemma states that, in every drawing of a graph with n vertices and m ≥ 4n edges there are at least Ω(m3/n2) pairs of crossing edges; or equivalently, there is an edge that crosses Ω(m2/n2) other edges. We strengthen the Crossing Lemma for drawings in which any two edges cross in at most O(1) points.

We prove for every \(k\in {\mathbb N}\) that every graph G with n vertices and m ≥ 3n edges drawn in the plane such that any two edges intersect in at most k points has two disjoint subsets of edges, E1 and E2, each of size at least \(c_km^2/n^2\), such that every edge in E1 crosses all edges in E2, where ck > 0 only depends on k. This bound is best possible up to the constant ck for every \(k\in {\mathbb N}\). We also prove that every graph G with n vertices and m ≥ 3n edges drawn in the plane with x-monotone edges has disjoint subsets of edges, E1 and E2, each of size Ω(m2/ (n2 polylog n)), such that every edge in E1 crosses all edges in E2. On the other hand, we construct x-monotone drawings of bipartite dense graphs where the largest such subsets E1 and E2 have size O(m2/(n2 log(m/n))).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Jacob Fox
    • 1
  • János Pach
    • 2
  • Csaba D. Tóth
    • 3
  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.City CollegeCUNY and Courant InstituteNYU, New York, NYUSA
  3. 3.University of CalgaryCalgaryCanada

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