منابع مشابه
Indecomposable Coverings
We prove that for every k > 1, there exist k-fold coverings of the plane (1) with strips, (2) with axis-parallel rectangles, and (3) with homothets of any fixed concave quadrilateral, that cannot be decomposed into two coverings. We also construct, for every k > 1, a set of points P and a family of disks D in the plane, each containing at least k elements of P , such that no matter how we color...
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We disprove the 1980 conjecture of János Pach about the cover-decomposability of open convex sets by showing that the unit disc is not cover-decomposable. In fact, our proof easily generalizes to any set with a smooth boundary. We also show that (the suitable variant of) the conjecture holds for unbounded sets. Let C be a collection of sets in R2. We say that C is an m-fold covering if every po...
متن کاملIndecomposable coverings with homothetic polygons
We prove that for any polygon S with at least four sides, and any k > 0, there is a k-fold covering of the plane with homothetic copies of S that can not be decomposed into two coverings.
متن کاملDecomposition tree and indecomposable coverings
Let G = (V,A) be a directed graph. With any subset X of V is associated the directed subgraph G[X ] = (X,A ∩ (X × X)) of G induced by X . A subset X of V is an interval of G provided that for a, b ∈ X and x ∈ V \ X , (a, x) ∈ A if and only if (b, x) ∈ A, This work was supported, in part, by an NSF EPSCoR grant EPS-0346476 and by a Nebraska Research Initiative (NRI) grant on high performance wir...
متن کاملIndecomposable Polytopes
The space of summands (with respect to vector addition) of a convex polytope in n dimensions is studied. This space is shown to be isomorphic to a convex pointed cone in Euclidean space. The extreme rays of this cone correspond to similarity classes of indecomposable polytopes. The decomposition of a polytope is described and a bound is given for the number of indecomposable summands needed. A ...
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ژورنال
عنوان ژورنال: Canadian Mathematical Bulletin
سال: 2009
ISSN: 0008-4395,1496-4287
DOI: 10.4153/cmb-2009-048-x