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Survey on decomposition of multiple coverings
 in Geometryintuitive, discrete, and convex, 219257, Bolyai Soc. Math. Stud., 24, János Bolyai Math. Soc
, 2013
"... The study of multiple coverings was initiated by Davenport and L. Fejes Tóth more than 50 years ago. In 1980 and 1986, the rst named author published the rst papers about decomposability of multiple coverings. It was discovered much later that, besides its theoretical interest, this area has practi ..."
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The study of multiple coverings was initiated by Davenport and L. Fejes Tóth more than 50 years ago. In 1980 and 1986, the rst named author published the rst papers about decomposability of multiple coverings. It was discovered much later that, besides its theoretical interest, this area has practical applications to sensor networks. Now there is a lot of activity in this eld with several breakthrough results, although, many basic questions are still unsolved. In this survey, we outline the most important results, methods, and questions. 1 Coverdecomposability and the sensor cover problem Let P = { Pi  i ∈ I} be a collection of sets in Rd. We say that P is an mfold covering if every point of Rd is contained in at least m members of P. The largest such m is called the thickness of the covering. A 1fold covering is simply called a covering. To formulate the central question of this survey succinctly, we need a denition. Denition 1.1. A planar set P is said to be coverdecomposable if there exists a (minimal) constant m = m(P) such that every mfold covering of the plane with translates of P can be decomposed into two coverings. Note that the above term is slightly misleading: we decompose (partition) not the set P, but a collection P of its translates. Such a partition is sometimes regarded a coloring of the members of P.
Unsplittable coverings in the plane
, 2015
"... A system of sets forms an mfold covering of a set X if every point of X belongs to at least m of its members. A 1fold covering is called a covering. The problem of splitting multiple coverings into several coverings was motivated by classical density estimates for sphere packings as well as by th ..."
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A system of sets forms an mfold covering of a set X if every point of X belongs to at least m of its members. A 1fold covering is called a covering. The problem of splitting multiple coverings into several coverings was motivated by classical density estimates for sphere packings as well as by the planar sensor cover problem. It has been the prevailing conjecture for 35 years (settled in many special cases) that for every plane convex body C, there exists a constant m =m(C) such that every mfold covering of the plane with translates of C splits into 2 coverings. In the present paper, it is proved that this conjecture is false for the unit disk. The proof can be generalized to construct, for every m, an unsplittable mfold covering of the plane with translates of any open convex body C which has a smooth boundary with everywhere positive curvature. Somewhat surprisingly, unbounded open convex sets C do not misbehave, they satisfy the conjecture: every 3fold covering of any region of the plane by translates of such a set C splits into two coverings. To establish this result, we prove a general coloring theorem for hypergraphs of a special type: shiftchains. We also show that there is a constant c> 0 such that, for any positive integer m, every mfold covering of a region with unit disks splits into two coverings, provided that every point is covered by at most c2m/2 sets.
MORE ON DECOMPOSING COVERINGS BY OCTANTS
 JOURNAL OF COMPUTATIONAL GEOMETRY
, 2015
"... In this note we improve our upper bound given in [7] by showing that every 9fold covering of a point set in R3 by finitely many translates of an octant decomposes into two coverings, and our lower bound by a construction for a 4fold covering that does not decompose into two coverings. The same bou ..."
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In this note we improve our upper bound given in [7] by showing that every 9fold covering of a point set in R3 by finitely many translates of an octant decomposes into two coverings, and our lower bound by a construction for a 4fold covering that does not decompose into two coverings. The same bounds also hold for coverings of points in R2 by finitely many homothets or translates of a triangle. We also prove that certain dynamic interval coloring problems are equivalent to the above question.