TY - JOUR

T1 - Coloring planar homothets and three-dimensional hypergraphs

AU - Cardinal, Jean

AU - Korman, Matias

PY - 2013/7/19

Y1 - 2013/7/19

N2 - 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.

AB - 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.

KW - 4-Color Theorem

KW - Chromatic numbers

KW - Conflict-free coloring

KW - Geometric hypergraphs

KW - Hypergraph coloring

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U2 - 10.1016/j.comgeo.2013.06.004

DO - 10.1016/j.comgeo.2013.06.004

M3 - Article

AN - SCOPUS:84880170725

VL - 46

SP - 1027

EP - 1035

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 9

ER -