Geometric Graphs with No Self-intersecting Path of Length Three

  • János Pach
  • Rom Pinchasi
  • Gábor Tardos
  • Géza Tóth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2528)

Abstract

Let G be a geometric graph with n vertices, i.e., a graph drawn in the plane with straight-line edges. It is shown that if G has no self-intersecting path of length 3, then its number of edges is O(n log n). This result is asymptotically tight. Analogous questions for curvilinear drawings and for longer paths are also considered.

References

  1. AAPPS97.
    P. K. Agarwal, B. Aronov, J. Pach, R. Pollack, and M. Sharir, Quasiplanargraphs have a linear number of edges, Combinatorica 17 (1997),1–9.MATHCrossRefMathSciNetGoogle Scholar
  2. AH66.
    S. Avital and H. Hanani, Graphs (in Hebrew), Gilyonot Lematematika 3 (1966), 2–8.Google Scholar
  3. B78.
    B. Bollobás, Extremal Graph Theory, Academic Press, New York, 1978.MATHGoogle Scholar
  4. BS74.
    A. Bondy and M. Simonovits, Cycles of even length in graphs, J. Combinatorial Theory, Ser. B 16 (1974), 97–105.MATHCrossRefMathSciNetGoogle Scholar
  5. F91.
    Z. Füredi, On a Turán type problem of Erdös, Combinatorica 11 1 (1991),75–79.CrossRefGoogle Scholar
  6. FH92.
    Z. Füredi and P. Hajnal, Davenport-Schinzel theory of matrices, Discrete Mathematics 103 (1992), 233–251.MATHCrossRefMathSciNetGoogle Scholar
  7. H34.
    H. Hanani (C. Chojnacki), Über wesentlich unplättbare Kurven in dreidimensionalen Raume, Fundamenta Mathematicae 23 (1934), 135–142.Google Scholar
  8. K79.
    Y. Kupitz, Extremal Problems in Combinatorial Geometry, Aarhus University Lecture Notes Series 53, Aarhus University, Denmark, 1979.Google Scholar
  9. P99.
    J. Pach, Geometric graph theory, in: Surveys in Combinatorics, 1999 (J. D. Lamb and D. A. Preece, eds.), London Mathematical Society Lecture Notes 267, Cambridge University Press, Cambridge, 1999, 167–200.Google Scholar
  10. PT97.
    J. Pach and G. Tóth, Graphs drawn with few crossings per edge, Combinatorica 17 (1997), 427–439.MATHCrossRefMathSciNetGoogle Scholar
  11. PR02.
    R. Pinchasi and R. Radoičić, On the number of edges in geometric graphswith no self-intersecting cycle of length 4, to appear.Google Scholar
  12. T02.
    G. Tardos, On the number of edges in a geometric graph with no shortself-intersecting paths, to appear.Google Scholar
  13. T70.
    W. T. Tutte, Toward a theory of crossing numbers, J. Combinatorial Theory 8 (1970), 45–53.MATHMathSciNetCrossRefGoogle Scholar
  14. V98.
    P. Valtr, On geometric graphs with no k pairwise parallel edges, Discrete and Computational Geometry 19 (1998), 461–469.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • János Pach
    • 1
  • Rom Pinchasi
    • 2
  • Gábor Tardos
    • 3
  • Géza Tóth
    • 4
  1. 1.City College, CUNY and Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  3. 3.Rényi Institute of the Hungarian Academy of SciencesHungary
  4. 4.Rényi Institute of the Hungarian Academy of SciencesHungary

Personalised recommendations