Recently, it was proved that triangle-free intersection graphs of n line
segments in the plane can have chromatic number as large as Θ(loglogn). Essentially the same construction produces Θ(loglogn)-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(loglogn), improving on
the previous bound of O(logn). 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(loglogn) colors using a combination of
techniques from on-line algorithms (first-fit) and data structure design
(heavy-light decomposition).Comment: Minor revisio