Construction of Locally Plane Graphs with Many Edges



A graph drawn in the plane with straight-line edges is called a geometric graph. If no path of length at most k in a geometric graph G is self-intersecting, we call Gk-locally plane. The main result of this chapter is a construction of k-locally plane graphs with a superlinear number of edges. For the proof, we develop randomized thinning procedures for edge-colored bipartite (abstract) graphs that can be applied to other problems as well.



The author’s research was partially supported by NSERC Grant 329527 and by OTKA Grants T-046234, AT048826, and NK-62321.


  1. 1.
    E. Ackerman, On the maximum number of edges in topological graphs with no four pairwise crossing edges. Discrete Comput. Geom. 41(3), 365–375 (2009)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    P. Agarwal, B. Aronov, J. Pach, R. Pollack, M. Sharir, Quasi-planar graphs have a linear number of edges. Combinatorica 17, 1–9 (1997)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    S. Avital, H. Hanani, Graphs. Gilyonot Lematematika 3, 2–8 (1966) [in Hebrew]Google Scholar
  4. 4.
    A. Marcus, G. Tardos, Intersection reverse sequences and geometric applications. J. Comb. Theor. Ser. A 113, 675–691 (2006)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    J. Pach, Geometric graph theory, in Surveys in Combinatorics, 1999, ed. by J. Lamb, D. Preece. London Mathematical Society Lecture Notes, vol. 267 (Cambridge University Press, Cambridge, 1999), pp. 167–200Google Scholar
  6. 6.
    J. Pach, Geometric graph theory, in Handbook of Discrete and Computational Geometry, 2nd edition, ed. by J. Goodman, J. O’Rourke (Chapman & Hall/CRC, Boca Raton, FL, 2004) ( Chapter 10)
  7. 7.
    J. Pach, R. Pinchasi, G. Tardos, G. Tóth, Geometric graphs with no self-intersecting path of length three. Eur. J. Combinator. 25(6), 793–811 (2004)MATHCrossRefGoogle Scholar
  8. 8.
    J. Pach, R. Radoičić, G. Tóth, Relaxing planarity for topological graphs, in Discrete and Computational Geometry, ed. by J. Akiyama, M. Kano. Lecture Notes in Computer Science, vol. 2866 (Springer, Berlin, 2003), pp. 221–232Google Scholar
  9. 9.
    J. Pach, G. Tóth, Graph drawn with few crossings per edge. Combinatorica 17, 427–439 (1997)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    R. Pinchasi, R. Radoičić, On the number of edges in geometric graphs with no self-intersecting cycle of length 4, in Towards a Theory of Geometric Graphs. Contemporary Mathematics, vol. 342 (American Mathematical Society, Providence, RI, 2004), pp. 233–243Google Scholar
  11. 11.
    P. Valtr, Graph drawing with no k pairwise crossing edges, in Graph Drawing (Rome, 1997). Lecture Notes in Computer Science, vol. 1353 (Springer-Verlag, Berlin, 1997), pp. 205–218Google Scholar
  12. 12.
    G. Tardos, On 0-1 matrices and small excluded submatrices. J. Combinator. Theor. Ser. A 111, 266–288 (2005)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    G. Tardos, G. Tóth, Crossing stars in topological graphs. SIAM J. Discrete Math. 21, 737–749 (2007)MathSciNetMATHCrossRefGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Rényi Institute, Budapest, Hungary; and School of Computing ScienceSimon Fraser UniversityBurnabyCanada

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