dc.contributor.author Ábrego, Bernardo M. dc.contributor.author Fernández Merchant, Silvia dc.contributor.author Kano, Mikio dc.contributor.author Orden, David dc.contributor.author Pérez Lantero, Pablo dc.contributor.author Seara Ojea, Carlos dc.contributor.author Tejel Altarriba, Francisco Javier dc.contributor.other Universitat Politècnica de Catalunya. Departament de Matemàtiques dc.date.accessioned 2020-01-30T12:29:41Z dc.date.available 2020-01-30T12:29:41Z dc.date.issued 2019-01-31 dc.identifier.citation Ábrego, B. [et al.]. K-1,K-3-covering red and blue points in the plane. "Discrete mathematics and theoretical computer science", 31 Gener 2019, vol. 21, núm. 3, p. 1-29. dc.identifier.issn 1462-7264 dc.identifier.other https://arxiv.org/pdf/1707.06856.pdf dc.identifier.uri http://hdl.handle.net/2117/176195 dc.description.abstract We say that a finite set of red and blue points in the plane in general position can be K1,3-covered if the set can be partitioned into subsets of size 4, with 3 points of one color and 1 point of the other color, in such a way that, if at each subset the fourth point is connected by straight-line segments to the same-colored points, then the resulting set of all segments has no crossings. We consider the following problem: Given a set R of r red points and a set B of b blue points in the plane in general position, how many points of R ¿ B can be K1,3-covered? and we prove the following results: (1) If r = 3g + h and b = 3h + g, for some non-negative integers g and h, then there are point sets R ¿ B, like {1, 3}-equitable sets (i.e., r = 3b or b = 3r) and linearly separable sets, that can be K1,3-covered. (2) If r = 3g + h, b = 3h + g and the points in R ¿ B are in convex position, then at least r + b - 4 points can be K1,3-covered, and this bound is tight. (3) There are arbitrarily large point sets R ¿ B in general position, with r = b + 1, such that at most r + b - 5 points can be K1,3-covered. (4) If b = r = 3b, then at least 8 9 (r + b - 8) points of R ¿ B can be K1,3-covered. For r > 3b, there are too many red points and at least r - 3b of them will remain uncovered in any K1,3-covering. dc.format.extent 29 p. dc.language.iso eng dc.publisher Chapman & Hall/CRC dc.subject Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica aplicada a les ciències dc.subject.lcsh Computer science--Mathematics dc.subject.other Non-crossing geometric graph dc.subject.other star dc.subject.other covering dc.subject.other red and blue points dc.title K-1,K-3-covering red and blue points in the plane dc.type Article dc.subject.lemac Informàtica--Matemàtica dc.contributor.group Universitat Politècnica de Catalunya. CGA - Computational Geometry and Applications dc.identifier.doi 10.23638/DMTCS-21-3-6 dc.description.peerreviewed Peer Reviewed dc.subject.ams Classificació AMS::68 Computer science::68R Discrete mathematics in relation to computer science dc.relation.publisherversion https://dmtcs.episciences.org/5126 dc.rights.access Open Access local.identifier.drac 25821550 dc.description.version Postprint (published version) dc.relation.projectid info:eu-repo/grantAgreement/MINECO//MTM2015-63791-R/ES/GRAFOS Y GEOMETRIA: INTERACCIONES Y APLICACIONES/ dc.relation.projectid info:eu-repo/grantAgreement/EC/H2020/734922/EU/Combinatorics of Networks and Computation/CONNECT local.citation.author Ábrego, B.; Fernández, S.; Kano, M.; Orden, D.; Perez-Lantero, P.; Seara, C.; Tejel, F. local.citation.publicationName Discrete mathematics and theoretical computer science local.citation.volume 21 local.citation.number 3 local.citation.startingPage 1 local.citation.endingPage 29
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