Crossing Number of Toroidal Graphs

  • János Pach
  • Géza Tóth
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3843)


It is shown that if a graph of n vertices can be drawn on the torus without edge crossings and the maximum degree of its vertices is at most d, then its planar crossing number cannot exceed cdn, where c is a constant. This bound, conjectured by Brass, cannot be improved, apart from the value of the constant. We strengthen and generalize this result to the case when the graph has a crossing-free drawing on an orientable surface of higher genus and there is no restriction on the degrees of the vertices.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • János Pach
    • 1
  • Géza Tóth
    • 2
  1. 1.City College, CUNY and Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Rényi InstituteHungarian Academy of SciencesBudapestHungary

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