Online and quasi-online colorings of wedges and intervals

TL;DR

This paper introduces online coloring algorithms for intervals and wedges that use logarithmic colors relative to the number of intervals and the parameter k, improving understanding of geometric hypergraph colorings.

Contribution

The paper presents new online coloring algorithms for intervals and wedges with bounds on the number of colors, contrasting with previous quasi-online results.

Findings

Online coloring of intervals uses Θ(log N/k) colors.

Online coloring of wedges achieves similar bounds.

Efficient algorithms are provided for all cases.

Abstract

We consider proper online colorings of hypergraphs defined by geometric regions. We prove that there is an online coloring algorithm that colors $N$ intervals of the real line using $\Theta(\log N/k)$ colors such that for every point $p$, contained in at least $k$ intervals, not all the intervals containing $p$ have the same color. We also prove the corresponding result about online coloring a family of wedges (quadrants) in the plane that are the translates of a given fixed wedge. These results contrast the results of the first and third author showing that in the quasi-online setting 12 colors are enough to color wedges (independent of $N$ and $k$). We also consider quasi-online coloring of intervals. In all cases we present efficient coloring algorithms.