Disjointness graphs of short polygonal chains

Abstract

The disjointness graph of a set system is a graph whose vertices are the sets, two being connected by an edge if and only if they are disjoint. It is known that the disjointness graph G of any system of segments in the plane is χ-bounded, that is, its chromatic number $\chi \left(G\right)$ is upper bounded by a function of its clique number $\omega \left(G\right)$.

Here we show that this statement does not remain true for systems of polygonal chains of length 2. We also construct systems of polygonal chains of length 3 such that their disjointness graphs have arbitrarily large girth and chromatic number. In the opposite direction, we show that the class of disjointness graphs of (possibly self-intersecting) 2-way infinite polygonal chains of length 3 is χ-bounded: for every such graph G, we have $\chi \left(G\right)\le {\left(\omega \right(G\left)\right)}^3+\omega \left(G\right)$.