, Volume 73, Issue 4, pp 673–695 | Cite as

On the Number of Edges of Fan-Crossing Free Graphs

  • Otfried CheongEmail author
  • Sariel Har-Peled
  • Heuna Kim
  • Hyo-Sil Kim


A graph drawn in the plane with \(n\) vertices is \(k\) -fan-crossing free for \(k \geqslant 2\) if there are no \(k+1\) edges \(g,e_1,\ldots , e_k\), such that \(e_1,e_2,\ldots ,e_k\) have a common endpoint and \(g\) crosses all \(e_i\). We prove a tight bound of \(4n-8\) on the maximum number of edges of a \(2\)-fan-crossing free graph, and a tight \(4n-9\) bound for a straight-edge drawing. For \(k \geqslant 3\), we prove an upper bound of \(3(k-1)(n-2)\) edges. We also discuss generalizations to monotone graph properties.


Graph drawing K-planar graphs Crossing number 



For helpful discussions, we thank David Eppstein, János Pach, Antoine Vigneron, Yoshio Okamoto, and Shin-ichi Tanigawa, as well as the other participants of the Korean Workshop on Computational Geometry 2011 in Otaru, Japan.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Otfried Cheong
    • 1
    Email author
  • Sariel Har-Peled
    • 2
  • Heuna Kim
    • 3
  • Hyo-Sil Kim
    • 4
  1. 1.Department of Computer ScienceKAISTDaejeonKorea
  2. 2.Department of Computer ScienceUniversity of IllinoisUrbanaUSA
  3. 3.Freie Universität BerlinBerlinGermany
  4. 4.Department of Computer Science and EngineeringPOSTECHPohangKorea

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