Covering the plane with convex polygons
Discrete & Computational Geometry
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
A simple algorithm for edge-coloring bipartite multigraphs
Information Processing Letters
Bounded size components: partitions and transversals
Journal of Combinatorial Theory Series B
Decomposition of multiple coverings into many parts
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Restricted strip covering and the sensor cover problem
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
On The Chromatic Number of Geometric Hypergraphs
SIAM Journal on Discrete Mathematics
Multiple Coverings of the Plane with Triangles
Discrete & Computational Geometry
Decomposition of multiple coverings into more parts
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Coloring Geometric Range Spaces
Discrete & Computational Geometry
A constructive proof of the Lovász local lemma
Proceedings of the forty-first annual ACM symposium on Theory of computing
CJCDGCGT'05 Proceedings of the 7th China-Japan conference on Discrete geometry, combinatorics and graph theory
Cover-decomposition and polychromatic numbers
ESA'11 Proceedings of the 19th European conference on Algorithms
Polychromatic coloring for half-planes
SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
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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(4ln 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. Lower bounds are also given for all of the above problems. 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.