Balanced partitions of 3-colored geometric sets in the plane

Sergey Bereg, Ferran Hurtado, Mikio Kano, Matias Korman, Dolores Lara, Carlos Seara, Rodrigo I. Silveira, Jorge Urrutia, Kevin Verbeek

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)

Abstract

Let S be a finite set of geometric objects partitioned into classes or colors. A subset S′⊆S is said to be balanced if S′ contains the same amount of elements of S from each of the colors. We study several problems on partitioning 3-colored sets of points and lines in the plane into two balanced subsets: (a) We prove that for every 3-colored arrangement of lines there exists a segment that intersects exactly one line of each color, and that when there are 2m lines of each color, there is a segment intercepting m lines of each color. (b) Given n red points, n blue points and n green points on any closed Jordan curve γ, we show that for every integer k with 0≤k≤n there is a pair of disjoint intervals on γ whose union contains exactly k points of each color. (c) Given a set S of n red points, n blue points and n green points in the integer lattice satisfying certain constraints, there exist two rays with common apex, one vertical and one horizontal, whose union splits the plane into two regions, each one containing a balanced subset of S.

Original languageEnglish
Pages (from-to)21-32
Number of pages12
JournalDiscrete Applied Mathematics
Volume181
DOIs
Publication statusPublished - 2015 Jan 30

Keywords

  • Bipartition
  • Colored point sets
  • Duality
  • Ham-sandwich theorem

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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