On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles
Discrete & Computational Geometry
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
The complexity of the union of (&agr;, &bgr;)-covered objects
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
On conflict-free coloring of points and simple regions in the plane
Proceedings of the nineteenth annual symposium on Computational geometry
Online conflict-free coloring for intervals
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Graphs and Hypergraphs
Conflict-free colorings of shallow discs
Proceedings of the twenty-second annual symposium on Computational geometry
Conflict-free coloring for intervals: from offline to online
Proceedings of the eighteenth annual ACM symposium on Parallelism in algorithms and architectures
Decomposition of multiple coverings into many parts
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Conflict-free coloring for rectangle ranges using O(n.382) colors
Proceedings of the nineteenth annual ACM symposium on Parallel algorithms and architectures
Weakening the online adversary just enough to get optimal conflict-free colorings for intervals
Proceedings of the nineteenth annual ACM symposium on Parallel algorithms and architectures
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
Delaunay graphs of point sets in the plane with respect to axis-parallel rectangles
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Coloring geometric range spaces
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
Notes: Coloring axis-parallel rectangles
Journal of Combinatorial Theory Series A
Conflict-Free colorings of rectangles ranges
STACS'06 Proceedings of the 23rd Annual conference on Theoretical Aspects of Computer Science
Online conflict-free colorings for hypergraphs
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
The potential to improve the choice: list conflict-free coloring for geometric hypergraphs
Proceedings of the twenty-seventh annual symposium on Computational geometry
Coloring planar homothets and three-dimensional hypergraphs
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
Coloring planar homothets and three-dimensional hypergraphs
Computational Geometry: Theory and Applications
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In FOCS 2002, Even et al. showed that any set of n discs in the plane can be Conflict-Free colored with a total of at most O(logn) colors. That is, it can be colored with O(logn) colors such that for any (covered) point p there is some disc whose color is distinct from all other colors of discs containing p. They also showed that this bound is asymptotically tight. In this paper we prove the following stronger results: (i) Any set of n discs in the plane can be colored with a total of at most O(k logn) colors such that (a) for any point p that is covered by at least k discs, there are at least k distinct discs each of which is colored by a color distinct from all other discs containing p and (b) for any point p covered by at most k discs, all discs covering p are colored distinctively. We call such a coloring a k-Strong Conflict-Free coloring. We extend this result to pseudo-discs and arbitrary regions with linear union-complexity. (ii) More generally, for families of n simple closed Jordan regions with union-complexity bounded by O(n1+α), we prove that there exists a k-Strong Conflict-Free coloring with at most O(knα) colors. (iii) We prove that any set of n axis-parallel rectangles can be k-Strong Conflict-Free colored with at most O(k log2n) colors. (iv) We provide a general framework for k-Strong Conflict-Free coloring arbitrary hypergraphs. This framework relates the notion of k-Strong Conflict-Free coloring and the recently studied notion of k-colorful coloring. All of our proofs are constructive. That is, there exist polynomial time algorithms for computing such colorings.