TY - JOUR

T1 - Balanced partitions of 3-colored geometric sets in the plane

AU - Bereg, Sergey

AU - Hurtado, Ferran

AU - Kano, Mikio

AU - Korman, Matias

AU - Lara, Dolores

AU - Seara, Carlos

AU - Silveira, Rodrigo I.

AU - Urrutia, Jorge

AU - Verbeek, Kevin

N1 - Funding Information:
F.H., M.K., C.S., and R.S. were partially supported by projects MINECO MTM2012-30951, Gen. Cat. DGR2009SGR1040 and DGR2014SGR46, and ESF EUROCORES programme EuroGIGA, CRP ComPoSe: MICINN Project EUI-EURC-2011-4306. F.H. and C.S. were partially supported by project MICINN MTM2009-07242. M.K. also acknowledges the support of the Secretary for Universities and Research of the Ministry of Economy and Knowledge of the Government of Catalonia and the European Union. R.I.S. was funded by Portuguese funds through CIDMA and FCT, within project PEst-OE/MAT/UI4106/2014, and by FCT grant SFRH/BPD/88455/2012 .

PY - 2015/1/30

Y1 - 2015/1/30

N2 - 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.

AB - 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.

KW - Bipartition

KW - Colored point sets

KW - Duality

KW - Ham-sandwich theorem

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U2 - 10.1016/j.dam.2014.10.015

DO - 10.1016/j.dam.2014.10.015

M3 - Article

AN - SCOPUS:84919481507

VL - 181

SP - 21

EP - 32

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

ER -