Coloring Hypergraphs Induced by Dynamic Point Sets and Bottomless Rectangles

Abstract

Abstract. We consider a coloring problem on dynamic, one-dimensional point sets: points appearing and disappearing on a line at given times. We wish to color them with k colors so that at any time, any sequence of p(k) consecutive points, for some function p, contains at least one point of each color.We prove that no such function p(k) exists in general. However, in the restricted case in which points appear gradually, but never disappear, we give a coloring algorithm guaranteeing the property at any time with p(k) = 3k − 2.This can be interpreted as coloring point sets in R 2 with k colors such that any bottomless rectangle containing at least 3k − 2 points contains at least one point of each color. Here a bottomless rectangle is an axis-aligned rectangle whose bottom edge is below the lowest point of the set.For this problem, we also prove a lower bound p(k) > ck, where c > 1.67. Hence for every k there exists a point set, every k-coloring of which is such that there exists a bottomless rectangle containing ck points and missing at least one of the k colors. Chen et al. (2009) proved that no such function p(k) exists in the case of general axis-aligned rectangles. Our result also complements recent results from Keszegh and Pálvölgyi on cover-decomposability of octants (2011, 2012).

An Abstract Approach to Polychromatic Coloring: Shallow Hitting Sets in ABA-free Hypergraphs and Pseudohalfplanes

“…Motivated by these, bottomless rectangles are regarded for small values in [11,12] and polychromatic k-colorings in [1]. In this paper we place bottomless rectangles in our abstract context and pose some further problems about them.…”

Making Octants Colorful and Related Covering Decomposition Problems

“…It can be expressed as coloring points appearing on a line in such a way that at all times any interval containing p(k) points contains one point of each color, or equivalently, coloring point sets in the plane such that every bottomless rectangle containing p(k) points contains a point of each color. In [5] it is shown that in this case a linear upper bound on p(k) can be achieved with a semi-online coloring algorithm or equivalently a sweeping line algorithm. 4…”

“…This setting can be thought of as points appearing on a line, and we want to color the points with k colors such that at any time, any set of p(k) consecutive points contains at least one of each color. This problem has been studied by a number of authors, whose results were compiled in a joint paper [5]. In particular, they showed that under this restriction, we have 1.6k p(k) 3k−2.…”

Making Octants Colorful and Related Covering Decomposition Problems

“…It can be expressed as coloring points appearing on a line in such a way that at all times any interval containing p(k) points contains one point of each color, or equivalently, coloring point sets in the plane such that every bottomless rectangle containing p(k) points contains a point of each color. In [5] it is shown that in this case a linear upper bound on p(k) can be achieved with a semi-online coloring algorithm or equivalently a sweeping line algorithm. 4 …”