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)


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.


Edge Incident Intersection Graph Absolute Constant Topological Graph Horizontal Strip 
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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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