Coloring curves that cross a fixed curve

Combinatorics Seminar
Friday, February 10, 2017 - 3:05pm
1 hour (actually 50 minutes)
Skiles 005
Jagiellonian University in Kraków
A class of graphs is *χ-bounded* if the chromatic number of all graphs in the class is bounded by some function of their clique number. *String graphs* are intersection graphs of curves in the plane. Significant research in combinatorial geometry has been devoted to understanding the classes of string graphs that are *χ*-bounded. In particular, it is known since 2012 that the class of all string graphs is not *χ*-bounded. We prove that for every integer *t*≥1, the class of intersection graphs of curves in the plane each of which crosses a fixed curve *c* in at least one and at most *t* points is *χ*-bounded. This result is best possible in several aspects; for example, the upper bound *t* on the number of crossings of each curve with *c* cannot be dropped. As a corollary, we obtain new upper bounds on the number of edges in so-called *k*-quasi-planar topological graphs. This is joint work with Alexandre Rok.