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Separated Matchings and Small Discrepancy Colorings

  • Viola Mészáros
Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7579)

Abstract

Consider 2n points in the plane in convex position, n points are red and n points are blue. Edges are straight line segments connecting points of different color. A separated matching is a geometrically non-crossing matching where all edges can be crossed by a line. Separated matchings are closely related to non-crossing, alternating paths. Abellanas et al. and independently Kynčl et al. constructed convex point sets allowing at most \(\frac{4}{3}n+O(\sqrt n)\) points on any non-crossing, alternating path. We give a class of configurations that contains at most \(\frac{4}{3}n+O(\sqrt n)\) points in any separated matching. We also present a coloring with constant discrepancy parameter where the number of points in the maximum separated matching is very close to \(\frac{4}{3}n\). When the dicrepancy is at most three we show that there are at least \(\frac{4}{3}n\) points in the maximum separated matching.

Keywords

Alternate Endpoint Hamiltonian Path Blue Point Minor Point Balance Coloring 
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.

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References

  1. 1.
    Abellanas, M., García, J., Hernández, G., Noy, M., Ramos, P.: Bipartite embeddings of trees in the plane. Discrete Appl. Math. 93, 141–148 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abellanas, M., García, A., Hurtado, H., Tejel, J.: Caminos Alternantes. In: Proc. X Encuentros de Geometría Computacional, Sevilla, pp. 7–12 (June 2003) (in Spanish)Google Scholar
  3. 3.
    Cibulka, J., Kynčl, J., Mészáros, V., Stolař, R., Valtr, P.: Hamiltonian Alternating Paths on Bicolored Double-Chains. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 181–192. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. 4.
    Cibulka, J., Kynčl, J., Mészáros, V., Stolař, R., Valtr, P.: Universal Sets for Straight-Line Embeddings of Bicolored-Graphs. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory. Springer (to appear)Google Scholar
  5. 5.
    Hajnal, P., Mészáros, V.: Note on noncrossing alternating path in colored convex sets. Accepted to Discrete Mathematics and Theoretical Computer ScienceGoogle Scholar
  6. 6.
    Kaneko, A., Kano, M.: Discrete geometry on red and blue points in the plane — a survey. In: Aronov, B., et al. (eds.) Discrete and Computational Geometry, Goodman-Pollack Festschrift. Algorithms Comb., vol. 25, pp. 551–570. Springer (2003)Google Scholar
  7. 7.
    Kynčl, J., Pach, J., Tóth, G.: Long alternating paths in bicolored point sets. Discrete Mathematics 308(19), 4315–4321 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Mészáros, V.: An upper bound on the size of separated matchings. Electronic Notes in Discrete Mathematics, 633–638 (2011)Google Scholar
  9. 9.
    Mészáros, V.: Separated matchings in colored convex sets (manuscript)Google Scholar
  10. 10.
    Mészáros, V.: Separated matchings in convex point sets with small discrepancy, CRM Documents 8, Centre de Recerca Matemática, Bellaterra, Barcelona, pp. 217–220 (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Viola Mészáros
    • 1
    • 2
  1. 1.Institute for MathematicsTU BerlinBerlinGermany
  2. 2.University of Szeged, Bolyai InstituteSzegedHungary

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