Coloring Hypergraphs Induced by Dynamic Point Sets and Bottomless Rectangles

TL;DR

This paper studies coloring dynamic point sets and bottomless rectangles to ensure diverse color coverage in certain geometric configurations, providing algorithms and bounds for specific cases.

Contribution

It introduces a coloring algorithm for gradually appearing points guaranteeing coverage in bottomless rectangles, and establishes bounds showing limitations in the general case.

Findings

A coloring algorithm guarantees coverage with p(k)=3k-2 for gradual point appearance.

A lower bound p(k)>ck (>1.67k) is proven for certain point sets.

No universal function p(k) exists for arbitrary dynamic point sets and rectangles.

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 Palvolgyi on cover-decomposability of octants (2011, 2012).