We prove that for every integer $$t\geqslant 1$$ , the class of intersection graphs of curves in the plane each of which crosses a fixed curve in at least one and at most t points is $$\chi $$ -bounded. This is essentially the strongest $$\chi $$ -boundedness result one can get for those kind of graph classes. As a corollary, we prove that for any fixed integers $$k\geqslant 2$$ and $$t\geqslant 1$$ , every k-quasi-planar topological graph on n vertices with any two edges crossing at most t times has $$O(n\log n)$$ edges.

An outerstring graph is an intersection graph of curves that lie in a common half-plane and have one endpoint on the boundary of that half-plane. We prove that the class of outerstring graphs is $\chi$-bounded, which means that their chromatic number is bounded by a function of their clique number. This generalizes a series of previous results on $\chi$-boundedness of outerstring graphs with various additional restrictions on the shape of curves or the number of times the pairs of curves can cross. The assumption that each curve has an endpoint on the boundary of the half-plane is justified by the known fact that triangle-free intersection graphs of straight-line segments can have arbitrarily large chromatic number.

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.