par Aloupis, Greg 
;Cardinal, Jean ;Collette, Sébastien ;Imahori, Shinji S.;Korman, Matias ;Langerman, Stefan ;Schwartz, Oded;Smorodinsky, Shakhar S.;Taslakian, Perouz
Référence Graphs and combinatorics, 27, 3
Publication Publié, 2010
;Cardinal, Jean ;Collette, Sébastien ;Imahori, Shinji S.;Korman, Matias ;Langerman, Stefan ;Schwartz, Oded;Smorodinsky, Shakhar S.;Taslakian, Perouz Référence Graphs and combinatorics, 27, 3
Publication Publié, 2010
Article révisé par les pairs
| Résumé : | We study the following geometric hypergraph coloring problem: given a planar point set and an integer k, we wish to color the points with k colors so that any axis-aligned strip containing sufficiently many points contains all colors. We show that if the strip contains at least 2k-1 points, such a coloring can always be found. In dimension d, we show that the same holds provided the strip contains at least k(4 ln k + ln d) points. We also consider the dual problem of coloring a given set of axis-aligned strips so that any sufficiently covered point in the plane is covered by k colors. We show that in dimension d the required coverage is at most d(k-1) + 1. This complements recent impossibility results on decomposition of strip coverings with arbitrary orientations. From the computational point of view, we show that deciding whether a three-dimensional point set can be 2-colored so that any strip containing at least three points contains both colors is NP-complete. This shows a big contrast with the planar case, for which this decision problem is easy. © 2011 Springer. |
