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Disjoint Edges in Topological Graphs and the Tangled-Thrackle Conjecture

  • Andres J. Ruiz-Vargas
  • Andrew Suk
  • Csaba D. Tóth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8871)

Abstract

It is shown that for a constant t ∈ ℕ, every simple topological graph on n vertices has O(n) edges if the graph has no two sets of t edges such that every edge in one set is disjoint from all edges of the other set (i.e., the complement of the intersection graph of the edges is K t,t -free). As an application, we settle the tangled-thrackle conjecture formulated by Pach, Radoičić, and Tóth: Every n-vertex graph drawn in the plane such that every pair of edges have precisely one point in common, where this point is either a common endpoint, a crossing, or a point of tangency, has at most O(n) edges.

Keywords

Edge Incident Intersection Graph Absolute Constant Topological Graph Horizontal Strip 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Andres J. Ruiz-Vargas
    • 1
  • Andrew Suk
    • 2
  • Csaba D. Tóth
    • 3
  1. 1.École polytechnique fédérale de LausanneLausanneSwitzerland
  2. 2.University of Illinois at ChicagoChicagoUSA
  3. 3.California State University NorthridgeLos AngelesUSA

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