# Disjointness graphs of segments

## TL;DR

This paper studies the properties of disjointness graphs of segments in space, establishing bounds on their chromatic and clique numbers, and explores computational complexity and coloring algorithms for these graphs.

## Contribution

It provides new bounds on the chromatic number of disjointness graphs of segments and lines, and shows NP-hardness of computing clique and chromatic numbers.

## Key findings

- Chromatic number bounded by a polynomial function of the clique number.
- NP-hardness of computing clique and chromatic numbers for line disjointness graphs.
- Existence of arc families with large chromatic number but small clique number.

## Abstract

The {\em disjointness graph} $G=G({\cal S})$ of a set of segments ${\cal S}$ in $R^d$, $d\ge 2,$ is a graph whose vertex set is ${\cal S}$ and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We prove that the chromatic number of $G$ satisfies $\chi(G)\le(\omega(G))^4+(\omega(G))^3$, where $\omega(G)$ denotes the clique number of $G$. It follows, that $\cal S$ has $\Omega(n^{1/5})$ pairwise intersecting or pairwise disjoint elements. Stronger bounds are established for lines in space, instead of segments. We show that computing $\omega(G)$ and $\chi(G)$ for disjointness graphs of lines in space are NP-hard tasks. However, we can design efficient algorithms to compute proper colorings of $G$ in which the number of colors satisfies the above upper bounds. One cannot expect similar results for sets of continuous arcs, instead of segments, even in the plane. We construct families of arcs whose disjointness graphs are triangle-free ($\omega(G)=2$), but whose chromatic numbers are arbitrarily large.