Covering the plane with convex polygons
Discrete & Computational Geometry
Decomposition of multiple coverings into many parts
Computational Geometry: Theory and Applications
Delaunay graphs of point sets in the plane with respect to axis-parallel rectangles
Random Structures & Algorithms - Proceedings of the Thirteenth International Conference “Random Structures and Algorithms” held May 28–June 1, 2007, Tel Aviv, Israel
Coloring Geometric Range Spaces
Discrete & Computational Geometry
Convex Polygons are Cover-Decomposable
Discrete & Computational Geometry
CJCDGCGT'05 Proceedings of the 7th China-Japan conference on Discrete geometry, combinatorics and graph theory
Notes: Coloring axis-parallel rectangles
Journal of Combinatorial Theory Series A
Indecomposable Coverings with Concave Polygons
Discrete & Computational Geometry
Decomposition of Multiple Coverings into More Parts
Discrete & Computational Geometry
Graphs and Combinatorics - The Japan Conference on Computational Geometry and Graphs (JCCGG2009)
Tight lower bounds for the size of epsilon-nets
Proceedings of the twenty-seventh annual symposium on Computational geometry
Optimally Decomposing Coverings with Translates of a Convex Polygon
Discrete & Computational Geometry
Polychromatic coloring for half-planes
Journal of Combinatorial Theory Series A
Octants Are Cover-Decomposable
Discrete & Computational Geometry
Octants are cover-decomposable into many coverings
Computational Geometry: Theory and Applications
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We consider a coloring problem on dynamic, one-dimensional point sets: points appearing and disappearing on a line at given times. We wish to color them with k colors so that at any time, any sequence of p(k) consecutive points, for some function p, contains at least one point of each color. We prove that no such function p(k) exists in general. However, in the restricted case in which points appear gradually, but never disappear, we give a coloring algorithm guaranteeing the property at any time with p(k)=3k−2. This can be interpreted as coloring point sets in ℝ2 with k colors such that any bottomless rectangle containing at least 3k−2 points contains at least one point of each color. Here a bottomless rectangle is an axis-aligned rectangle whose bottom edge is below the lowest point of the set. For this problem, we also prove a lower bound p(k)ck, where c1.67. Hence, for every k there exists a point set, every k-coloring of which is such that there exists a bottomless rectangle containing ck points and missing at least one of the k colors. Chen et al. (2009) proved that no such function p(k) exists in the case of general axis-aligned rectangles. Our result also complements recent results from Keszegh and Pálvölgyi on cover-decomposability of octants (2011, 2012).