Outerstring graphs are χ-bounded
Alexandre Rok, Bartosz Walczak · 2014
An outerstring graph is an intersection graph of curves lying in a halfplane with one endpoint on the boundary of the halfplane. It is proved that the outerstring graphs are χ-bounded, that is, their chromatic number is bounded by a function of their clique number. This generalizes a series of previous results on χ-boundedness of outerstring graphs with various restrictions of the shape of the curves or the number of times the pairs of curves can intersect. This also implies that the intersection graphs of x-monotone curves with bounded clique number have chromatic number O(log n), improving the previous polylogarithmic upper bound. The assumption that each curve has an endpoint on the boundary of the halfplane is justified by the known fact that triangle-free intersection graphs of straight-line segments can have arbitrarily large chromatic number.