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Relaxing Planarity for Topological Graphs

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Part of the Bolyai Society Mathematical Studies book series (BSMS,volume 15)

Abstract

According to Euler’s formula, every planar graph with n vertices has at most O(n) edges. How much can we relax the condition of planarity without violating the conclusion? After surveying some classical and recent results of this kind, we prove that every graph of n vertices, which can be drawn in the plane without three pairwise crossing edges, has at most O(n) edges. For straight-line drawings, this statement has been established by Agarwal et al., using a more complicated argument, but for the general case previously no bound better than O(n 3/2) was known.

Keywords

  • Chromatic Number
  • Topological Graph
  • Geometric Graph
  • Disjoint Edge
  • Bisection Width

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.

János Pach has been supported by NSF Grant CCR-00-98245, by PSC-CUNY Research Award 63352-0036, and by OTKA T-032458. Géza Tóth has been supported by OTKA-T-038397 and by an award from the New York University Research Challenge Fund.

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© 2006 János Bolyai Mathematical Society and Springer Verlag

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Pach, J., Radoičić, R., Tóth, G. (2006). Relaxing Planarity for Topological Graphs. In: Győri, E., Katona, G.O.H., Lovász, L., Fleiner, T. (eds) More Sets, Graphs and Numbers. Bolyai Society Mathematical Studies, vol 15. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-32439-3_12

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