On the Density of Non-simple 3-Planar Graphs

  • Michael A. BekosEmail author
  • Michael Kaufmann
  • Chrysanthi N. Raftopoulou
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9801)


A k-planar graph is a graph that can be drawn in the plane such that every edge is crossed at most k times. For \(k \le 4\), Pach and Tóth [20] proved a bound of \((k+3)(n-2)\) on the total number of edges of a k-planar graph, which is tight for \(k=1,2\). For \(k=3\), the bound of \(6n-12\) has been improved to \(\frac{11}{2}n-11\) in [19] and has been shown to be optimal up to an additive constant for simple graphs. In this paper, we prove that the bound of \(\frac{11}{2}n-11\) edges also holds for non-simple 3-planar graphs that admit drawings in which non-homotopic parallel edges and self-loops are allowed. Based on this result, a characterization of optimal 3-planar graphs (that is, 3-planar graphs with n vertices and exactly \(\frac{11}{2}n-11\) edges) might be possible, as to the best of our knowledge the densest known simple 3-planar is not known to be optimal.


Middle Part Jordan Curve Simple Graph Parallel Edge Free Graph 
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.



We thank E. Ackerman for bringing to our attention [1] and [19].


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Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Michael A. Bekos
    • 1
    Email author
  • Michael Kaufmann
    • 1
  • Chrysanthi N. Raftopoulou
    • 2
  1. 1.Institut für InformatikUniversität TübingenTubingenGermany
  2. 2.School of Applied Mathematics and Physical SciencesNTUAAthensGreece

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