How to net a lot with little: small &egr;-nets for disks and halfspaces
SCG '90 Proceedings of the sixth annual symposium on 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 kk-free intersection graphs of geometric objects in the plane
Proceedings of the twenty-fourth annual symposium on Computational geometry
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
Octants are cover-decomposable into many coverings
Computational Geometry: Theory and Applications
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We prove that every finite set of homothetic copies of a given compact and 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 obtain constant factor improvements for conflict-free coloring, k-strong conflict-free coloring, and choosability. Proofs of the upper bounds are constructive and yield simple, polynomial-time algorithms.