Coloring curves that cross a fixed curve

TL;DR

This paper proves that intersection graphs of curves crossing a fixed curve a limited number of times are chi-bounded, and applies this to bound edges in certain quasi-planar graphs, advancing understanding of graph coloring and crossing constraints.

Contribution

It establishes the chi-boundedness of a new class of intersection graphs and derives bounds on edges in quasi-planar graphs with limited crossings.

Findings

Intersection graphs of curves crossing a fixed curve are chi-bounded.

Every k-quasi-planar topological graph with limited crossings has O(n log n) edges.

Abstract

We prove that for every integer $t\geq 1$, 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 this kind of graph classes. As a corollary, we prove that for any fixed integers $k\geq 2$ and $t\geq 1$, every $k$-quasi-planar topological graph on $n$ vertices with any two edges crossing at most $t$ times has $O(n\log n)$ edges.