Coloring triangle-free rectangle overlap graphs with $O(\log\log n)$ colors

TL;DR

This paper proves that triangle-free rectangle overlap graphs can be colored with $O(\log\log n)$ colors, establishing an asymptotically optimal bound and improving previous results.

Contribution

It introduces a new bound of $O(\log\log n)$ for coloring triangle-free rectangle overlap graphs, matching the known lower bound and advancing understanding of geometric graph coloring.

Findings

Triangle-free rectangle overlap graphs have chromatic number $O(\log\log n)$.

The construction for other shapes is asymptotically optimal for rectangles.

A novel coloring approach combines online algorithms and data structures.

Abstract

Recently, it was proved that triangle-free intersection graphs of $n$ line segments in the plane can have chromatic number as large as $\Theta(\log\log n)$. Essentially the same construction produces $\Theta(\log\log n)$-chromatic triangle-free intersection graphs of a variety of other geometric shapes---those belonging to any class of compact arc-connected sets in $\mathbb{R}^2$ closed under horizontal scaling, vertical scaling, and translation, except for axis-parallel rectangles. We show that this construction is asymptotically optimal for intersection graphs of boundaries of axis-parallel rectangles, which can be alternatively described as overlap graphs of axis-parallel rectangles. That is, we prove that triangle-free rectangle overlap graphs have chromatic number $O(\log\log n)$, improving on the previous bound of $O(\log n)$. To this end, we exploit a relationship between off-line coloring of rectangle overlap graphs and on-line coloring of interval overlap graphs. Our coloring method decomposes the graph into a bounded number of subgraphs with a tree-like structure that "encodes" strategies of the adversary in the on-line coloring problem. Then, these subgraphs are colored with $O(\log\log n)$ colors using a combination of techniques from on-line algorithms (first-fit) and data structure design (heavy-light decomposition).