Triangle-free geometric intersection graphs with large chromatic number

TL;DR

The paper constructs large chromatic number intersection graphs of geometric objects in the plane that are triangle-free, answering longstanding questions and revealing connections to interval coloring.

Contribution

It introduces a general method to create triangle-free geometric intersection graphs with arbitrarily large chromatic number, including for shapes like circles and L-shapes.

Findings

Constructed triangle-free intersection graphs with large chromatic number for various shapes.

Provided a negative answer to Gyarfas and Lehel's question for L-shapes.

Linked planar geometric coloring problems to online interval coloring.

Abstract

Several classical constructions illustrate the fact that the chromatic number of a graph can be arbitrarily large compared to its clique number. However, until very recently, no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set $X$ in $\mathbb{R}^2$ that is not an axis-aligned rectangle and for any positive integer $k$ produces a family $\mathcal{F}$ of sets, each obtained by an independent horizontal and vertical scaling and translation of $X$, such that no three sets in $\mathcal{F}$ pairwise intersect and $\chi(\mathcal{F})>k$. This provides a negative answer to a question of Gyarfas and Lehel for L-shapes. With extra conditions, we also show how to construct a triangle-free family of homothetic (uniformly scaled) copies of a set with arbitrarily large chromatic number. This applies to many common shapes, like circles, square boundaries, and equilateral L-shapes. Additionally, we reveal a surprising connection between coloring geometric objects in the plane and on-line coloring of intervals on the line.