Separated Matchings and Small Discrepancy Colorings

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


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.


Alternate Endpoint Hamiltonian Path Blue Point Minor Point Balance Coloring 
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  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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