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Making Octants Colorful and Related Covering Decomposition Problems
 SIAM J. Discrete Math
, 1948
"... Abstract. We give new positive results on the longstanding open problem of geometric covering decomposition for homothetic polygons. In particular, we prove that for any positive integer k, every finite set of points in R3 can be colored with k colors so that every translate of the negative octant ..."
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Abstract. We give new positive results on the longstanding open problem of geometric covering decomposition for homothetic polygons. In particular, we prove that for any positive integer k, every finite set of points in R3 can be colored with k colors so that every translate of the negative octant containing at least k6 points contains at least one of each color. The best previously known bound was doubly exponential in k. This yields, among other corollaries, the first polynomial bound for the decomposability of multiple coverings by homothetic triangles. We also investigate related decomposition problems involving intervals appearing on a line. We prove that no algorithm can dynamically maintain a decomposition of a multiple covering by intervals under insertion of new intervals, even in a semionline model, in which some coloring decisions can be delayed. This implies that a wide range of sweeping plane algorithms cannot guarantee any bound even for special cases of the octant problem.
On ConflictFree MultiColoring?
"... Abstract A conflictfree coloring of a hypergraph H = (V, E), E ⊆ 2V, is a coloring of the vertices V such that every hyperedge E ∈ E contains a vertex of “unique ” color. Our goal is to minimize the total number of distinct colors. In its full generality, this problem is known as the conflictfree ..."
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Abstract A conflictfree coloring of a hypergraph H = (V, E), E ⊆ 2V, is a coloring of the vertices V such that every hyperedge E ∈ E contains a vertex of “unique ” color. Our goal is to minimize the total number of distinct colors. In its full generality, this problem is known as the conflictfree (hypergraph) coloring problem. It is known that Θ( m) colors might be needed in general. In this paper we study the relaxation of the problem where one is allowed to assign multiple colors to the same node. The goal here is to substantially reduce the total number of colors, while keeping the number of colors per node as small as possible. By a simple adaptation of a result by Pach and Tardos [2009] on the singlecolor version of the problem, one obtains that only O(log2m) colors in total are sufficient (on every instance) if each node is allowed to use up to O(logm) colors. By improving on the result of Pach and Tardos (under the assumption n m), we show that the same result can be achieved with O(logm·logn) colors in total, and eitherO(logm) orO(logn·log logm) ⊆ O(log2 n) colors per node. The latter coloring can be computed by a polynomialtime Las Vegas algorithm. 1
BROOKS TYPE RESULTS FOR CONFLICTFREE COLORINGS AND {a, b}FACTORS IN GRAPHS.
"... Abstract. A vertexcoloring of a hypergraph is conflictfree, if each edge contains a vertex whose color is not repeated on any other vertex of that edge. Let f(r,∆) be the smallest integer k such that each runiform hypergraph of maximum vertex degree ∆ has a conflictfree coloring with at most k c ..."
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Abstract. A vertexcoloring of a hypergraph is conflictfree, if each edge contains a vertex whose color is not repeated on any other vertex of that edge. Let f(r,∆) be the smallest integer k such that each runiform hypergraph of maximum vertex degree ∆ has a conflictfree coloring with at most k colors. As shown by Tardos and Pach, similarly to a classical Brooks ’ type theorem for hypergraphs, f(r,∆) ≤ ∆+1. Compared to Brooks’ theorem, according to which there is only a couple of graphs/hypergraphs that attain the ∆+1 bound, we show that there are several infinite classes of uniform hypergraphs for which the upper bound is attained. We give better upper bounds in terms of ∆ for large ∆ and establish the connection between conflictfree colorings and socalled {t, r−t}factors in rregular graphs. Here, a {t, r − t}factor is a factor in which each degree is either t or r − t. Among others, we disprove a conjecture of Akbari and Kano [1] stating that there is a {t, r − t}factor in every rregular graph for odd r and any odd t < r 3