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

Recently, Pawlik et al. have shown that triangle-free intersection graphs of line segments in the plane can have arbitrarily large chromatic number. Specifically, they construct triangle-free segment intersection graphs with chromatic number Θ(log log n). Essentially the same construction produces Θ(log log n)-chromatic triangle-free intersection graphs of a variety of other geometric shapes—those belonging to any class of compact arc-connected subsets of R 2 closed under horizontal scaling, vertical scaling, and translation, except for axis-aligned rectangles. We show that this construction is asymptotically optimal for intersection graphs of boundaries of axis-aligned rectangles. This class of graphs can be alternatively defined as the class of overlap graphs of axis-aligned rectangles, that is, graphs in which two rectangles are connected by an edge if they intersect but are not nested. 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, and colors these subgraphs with O(log log n) colors using a combination of techniques from on-line algorithms (first-fit) and data structure design (heavy-light decomposition).