Every planar graph is 5-choosable
Journal of Combinatorial Theory Series B
Efficiently four-coloring planar graphs
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Voronoi diagrams and Delaunay triangulations
Handbook of discrete and computational geometry
Conflict-Free Coloring of Points and Simple Regions in the Plane
Discrete & Computational Geometry
Decomposition of multiple coverings into many parts
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
On The Chromatic Number of Geometric Hypergraphs
SIAM Journal on Discrete Mathematics
Coloring Geometric Range Spaces
Discrete & Computational Geometry
Decomposing Coverings and the Planar Sensor Cover Problem
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
CJCDGCGT'05 Proceedings of the 7th China-Japan conference on Discrete geometry, combinatorics and graph theory
Conflict-Free coloring made stronger
SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
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We prove that every finite set of homothetic copies of a given convex body in the plane can be colored with four colors so that any point covered by at least two copies is covered by two copies with distinct colors. This generalizes a previous result from Smorodinsky (SIAM J. Disc. Math. 2007). Then we show that for any k=2, every three-dimensional hypergraph can be colored with 6(k-1) colors so that every hyperedge e contains min{|e|,k} vertices with mutually distinct colors. This refines a previous result from Aloupis et al. (Disc. & Comp. Geom. 2009). As corollaries, we improve on previous results for conflict-free coloring, k-strong conflict-free coloring, and choosability. Proofs of the upper bounds are constructive and yield simple, polynomial-time algorithms.