New Bounds on the Maximum Number of Edges in k-Quasi-Planar Graphs

  • Andrew Suk
  • Bartosz Walczak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8242)

Abstract

A topological graph is k-quasi-planar if it does not contain k pairwise crossing edges. An old conjecture states that for every fixed k, the maximum number of edges in a k-quasi-planar graph on n vertices is O(n). Fox and Pach showed that every k-quasi-planar graph with n vertices and no pair of edges intersecting in more than O(1) points has at most n(logn) O(logk) edges. We improve this upper bound to \(2^{\alpha(n)^c}n\log n\), where α(n) denotes the inverse Ackermann function, and c depends only on k. We also show that every k-quasi-planar graph with n vertices and every two edges have at most one point in common has at most O(nlogn) edges. This improves the previously known upper bound of \(2^{\alpha(n)^c}n\log n\) obtained by Fox, Pach, and Suk.

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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Andrew Suk
    • 1
  • Bartosz Walczak
    • 2
    • 3
  1. 1.Massachusetts Institute of TechnologyCambridgeUSA
  2. 2.École Polytechnique Fédérale de LausanneSwitzerland
  3. 3.Theoretical Computer Science Department, Faculty of Mathematics and Computer ScienceJagiellonian UniversityKrakówPoland

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