Long Alternating Paths in Bicolored Point Sets

  • Jan Kynčl
  • János Pach
  • Géza Tóth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3383)

Abstract

Given n red and n blue points in convex position in the plane, we show that there exists a noncrossing alternating path of length \(n+c\sqrt{n\over \log n}\). We disprove a conjecture of Erdős by constructing an example without any such path of length greater than \({4\over 3}n+c'\sqrt{n}\).

References

  1. [AGH97]
    Abellanas, M., Garcia, J., Hernández, G., Noy, M., Ramos, P.: Bipartite embeddings of trees in the plane. In: North, S.C. (ed.) GD 1996. LNCS, vol. 1190, pp. 141–148. Springer, Heidelberg (1997); Also in: Discrete Appl. Math. 93, 141–148 (1999)Google Scholar
  2. [AGHT03]
    Abellanas, M., Garcia, A., Hurtado, F., Tejel, J.: Caminos alternantes. In: X Encuentros de Geometría Computacional, Sevilla, pp. 7–12 (2003) (in Spanish) Google Scholar
  3. [BM01]
    Bárány, I., Matoušek, J.: Simultaneous partitions of measures by k-fans. Discrete Comput. Geom. 25, 317–334 (2001)MATHGoogle Scholar
  4. [BKS00]
    Bespamyatnikh, S., Kirkpatrick, D., Snoeyink, J.: Generalizing ham sandwich cuts to equitable subdivisions. Discrete Comput. Geom. 24, 605–622 (2000)MATHMathSciNetGoogle Scholar
  5. [dFPP90]
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10, 41–51 (1990)MATHCrossRefMathSciNetGoogle Scholar
  6. [CK89]
    Chrobak, M., Karloff, H.: A lower bound on the size of universal sets for planar graphs. SIGACT News 20/4, 63–86 (1989) Google Scholar
  7. [GMPP91]
    Gritzmann, P., Mohar, B., Pach, J., Pollack, R.: Embedding a planar triangulation with vertices at specified points (solution to Problem E3341). Amer. Math. Monthly 98, 165–166 (1991) Google Scholar
  8. [IPTT94]
    Ikeba, Y., Perles, M., Tamura, A., Tokunaga, S.: The rooted tree embedding problem into points on the plane. Discrete Comput. Geom. 11, 51–63 (1994)CrossRefMathSciNetGoogle Scholar
  9. [KK04]
    Kaneko, A., Kano, M.: Discrete geometry on red and blue poins in the plane – a survey. In: Aronov, B., et al. (eds.) Discrete and Computational Geometry, pp. 551–570. Springer, Berlin (2004)Google Scholar
  10. [KKS04]
    Kaneko, A., Kano, M., Suzuki, K.: Path coverings of two sets of points in the plane. In: Pach, J. (ed.) Towards a Theory of Geometric Graphs. Contemporary Mathematics, vol. 342, pp. 99–111 (2004) Google Scholar
  11. [KKY00]
    Kaneko, A., Kano, M., Yoshimoto, K.: Alternating Hamiltonian cycles with minimum number of crossings in the plane. Internat. J. Comput. Geom. Appl. 10, 73–78 (2000)MATHMathSciNetGoogle Scholar
  12. [KPTV98]
    Károlyi, G., Pach, J., Tóth, G., Valtr, P.: Ramsey-type results for geometric graphs II. Discrete Comput. Geom. 20, 375–388 (1998)MATHCrossRefMathSciNetGoogle Scholar
  13. [MSU05]
    Merino, C., Salazar, G., Urrutia, J.: On the length of longest alternating paths for multicolored point sets in convex position (manuscript) Google Scholar
  14. [S02]
    Sakai, T.: Balanced convex partitions of measures in R 2. Graphs and Combinatorics 18, 169–192 (2002)MATHCrossRefMathSciNetGoogle Scholar
  15. [T96]
    Tokunaga, S.: On a straight-line embedding problem of graphs. Discrete Math. 150, 371–378 (1996)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Jan Kynčl
    • 1
  • János Pach
    • 2
  • Géza Tóth
    • 3
  1. 1.Department of Applied Mathematics and, Institute for Theoretical Computer ScienceCharles UniversityPragueCzech Republic
  2. 2.City College, CUNY and Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  3. 3.Rényi Institute, Hungarian Academy of SciencesBudapestHungary

Personalised recommendations