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Spanning Trees of Multicoloured Point Sets with Few Intersections

  • J. Leaños
  • C. Merino
  • G. Salazar
  • J. Urrutia
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3330)

Abstract

Kano et al. proved that if P 0, P 1, ..., P k − − 1 are pairwise disjoint collections of points in general position, then there exist spanning trees T 0, T 1, ..., T k − − 1, of P 0, P 1, ..., P k − − 1, respectively, such that the edges of T 0 ∪ T 1 ∪ ... ∪ T k − 1 intersect in at most (k – 1)nk(k – 1)/2 points. In this paper we show that this result is asymptotically tight within a factor of 3/2. To prove this, we consider alternating collections, that is, collections such that the points in P: = P 0 ∪ P 1 ∪ ... ∪ P k − 1 are in convex position, and the points of the P i ’s alternate in the convex hull of P.

Keywords

Convex Hull Span Tree General Position Inductive Step Good Collection 
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References

  1. 1.
    Kaneko, A., Kano, M., Suzuki, K., Tokunaga, S.: Crossing Numbers of Three Monochromatic Trees in the Plane (preprint)Google Scholar
  2. 2.
    Kano, M., Merino, C., Urrutia, J.: On spanning trees and cycles of multicoloured point sets with few intersections (submitted)Google Scholar
  3. 3.
    Tokunaga, S.: Intersection number of two connected geometric graphs. Inform. Process. Lett. 59(6), 331–333 (1996)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • J. Leaños
    • 1
  • C. Merino
    • 2
  • G. Salazar
    • 3
  • J. Urrutia
    • 2
  1. 1.Facultad de CienciasUASLPSan Luis PotosíMexico
  2. 2.Instituto de MatemáticasUNAMMexico
  3. 3.Instituto de Investigación en Comunicación ÓpticaUASLPSan Luis PotosíMexico

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