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

Original · EN

Recently, it was proved that triangle-free intersection graphs of n line segments in the plane can have chromatic number as large as Θ(n). Essentially the same construction produces Θ(n)-chromatic triangle-free intersection graphs of a variety of other geometric shapes---those belonging to any class of compact arc-connected 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(n), improving on the previous bound of O(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(n) colors using a combination of techniques from on-line algorithms (first-fit) and data structure design (heavy-light decomposition).