Density Theorems for Intersection Graphs of t-Monotone Curves

  • Andrew Suk
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7704)


A curve γ in the plane is t-monotone if its interior has at most t − 1 vertical tangent points. A family of t-monotone curves F is simple if any two members intersect at most once. It is shown that if F is a simple family of n t-monotone curves with at least εn 2 intersecting pairs (disjoint pairs), then there exists two subfamilies F 1,F 2 ⊂ F of size δn each, such that every curve in F 1 intersects (is disjoint to) every curve in F 2, where δ depends only on ε. We apply these results to find pairwise disjoint edges in simple topological graphs.


Intersection Graph Density Theorem Topological Graph Left Endpoint Disjoint Edge 
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.


  1. 1.
    Agarwal, P.K., van Kreveld, M., Suri, S.: Label placement by maximum independent set in rectangles. Comput. Geom. Theory Appl. 11, 209–218 (1998)zbMATHCrossRefGoogle Scholar
  2. 2.
    Alon, N., Pach, J., Pinchasi, R., Radoicic, R., Sharir, M.: Crossing patterns of semi-algebraic sets. J. Comb. Theory Ser. A 111, 310–326 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Asano, T., Imai, H.: Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane. J. Algorithms 4, 310–323 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Asplund, E., Grünbaum, B.: On a coloring problem. Math. Scand. 8, 181–188 (1960)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Basu, S.: Combinatorial complexity in o-minimal geometry. Proc. London Math. Soc. 100, 405–428 (2010)zbMATHCrossRefGoogle Scholar
  6. 6.
    Cairns, G., Nikolayevsky, Y.: Bounds for generalized thrackles. Discrete Comput. Geom. 23, 191–206 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Ehrlich, G., Even, S., Tarjan, R.E.: Intersection graphs of curves in the plane. J. Combinatorial Theory, Ser. B 21, 8–20 (1979)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Erdős, P., Hajnal, A.: Ramsey-type theorems. Discrete Appl. Math. 25, 37–52 (1989)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Fox, J., Pach, J., Tóth: Intersection patterns of curves. Journal of the London Mathematical Society 83, 389–406 (2011)zbMATHCrossRefGoogle Scholar
  10. 10.
    Fox, J., Sudakov, B.: Density theorems for bipartite graphs and related Ramsey-type results. Combinatorica 29, 153–196 (2009)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Fulek, R., Pach, J.: A computational approach to Conway’s thrackle conjecture. Comput. Geom. 44, 345–355 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)zbMATHGoogle Scholar
  13. 13.
    Gyárfás, A.: On the chromatic number of multiple intervals graphs and overlap graphs. Discrete Math. 55, 161–166 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32, 130–136 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Kozik, J., Krawczyk, T., Lasoń, M., Micek, P., Pawlik, A., Trotter, W., Walczak, B.: Triangle-free intersection graphs of segments in the plane with arbitrarily large chromatic number (submitted)Google Scholar
  16. 16.
    Lovász, L., Pach, J., Szegedy, M.: On Conway’s thrackle conjecture. Discrete Comput. Geom. 18, 369–376 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Matoušek, J.: Lectures on Discrete Geometry. Springer, New York (2002)zbMATHCrossRefGoogle Scholar
  18. 18.
    Pach, J., Solymosi, J.: Crossing patterns of segments. J. Comb. Theory Ser. A 96, 316–325 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Pach, J., Sterling, E.: Conway’s conjecture for monotone thrackles. Amer. Math. Monthley 118, 544–548 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Pach, J., Töröcsik, J.: Some geometric applications of Dilworth’s theorem. Discrete Comput. Geom. 12, 1–7 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Pach, J., Tóth, G.: Which crossing number is it anyway? J. Comb. Theory Ser. B 80, 225–246 (2000)zbMATHCrossRefGoogle Scholar
  22. 22.
    Pach, J., Tóth, G.: Disjoint Edges in Topological Graphs. In: Akiyama, J., Baskoro, E.T., Kano, M. (eds.) IJCCGGT 2003. LNCS, vol. 3330, pp. 133–140. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  23. 23.
    Suk, A.: Coloring intersection graphs of x-monotone curves in the plane (submitted)Google Scholar
  24. 24.
    Suk, A.: Disjoint edges in complete topological graphs (to appear)Google Scholar
  25. 25.
    Tóth, G.: Note on geometric graphs. J. Comb. Theory Ser. A 89, 126–132 (2000)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Andrew Suk
    • 1
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations