Flipping Triangles and Rectangles

For any graph G, it is obvious that χ(G) ≥ ω(G), but whether χ(G) can be bounded from above by a function of ω(G) is not at all clear. Recall that the two numbers are equal for every induced subgraph of a perfect graph. Thus the ratio of the chromatic number to the clique number of a graph is often studied as a measure of its “imperfection”; see for example Dumitrescu and Jiang (2010) for some related work on bounding this ratio for geometric intersection graphs and their complements. Corollary 1. 1/2 ≤ limn→∞ (