# Coloring Triangle-Free Rectangle Overlap Graphs with $O(\log \log n)$ Colors

Discrete & Computational Geometry, Nov 2014

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).

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs00454-014-9640-3.pdf

Tomasz Krawczyk, Arkadiusz Pawlik, Bartosz Walczak. Coloring Triangle-Free Rectangle Overlap Graphs with $O(\log \log n)$ Colors, Discrete & Computational Geometry, 2015, 199-220, DOI: 10.1007/s00454-014-9640-3