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

Recently, it was proved that triangle-free intersection graphs of n line segments in the plane can have chromatic number as large as (log log n). Essentially the same construction produces (log log n)-chromatic triangle-free intersection graphs of a variety of other geometric shapes—those belonging to any class of compact arcconnected sets in R2 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 Preliminary version of this paper appeared as: Coloring triangle-free rectangular frame intersection graphs with O(log log n) colors. In: Brandstädt, A., Jansen, K., Reischuk, R. (eds.), Graph-Theoretic Concepts in Computer Science (WG 2013). Lecture Notes in Computer Science, vol. 8165, pp. 333–344. Springer, Berlin (2013). T. Krawczyk · A. Pawlik · B. Walczak Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland e-mail: [email protected] T. Krawczyk e-mail: [email protected] A. Pawlik e-mail: [email protected]