Box and segment intersection graphs with large girth and chromatic number

TL;DR

This paper demonstrates the existence of intersection graphs in three-dimensional space, specifically of axis-aligned boxes and straight lines, that can have arbitrarily large girth and chromatic number, highlighting complex coloring properties.

Contribution

It establishes the existence of intersection graphs with large girth and chromatic number in three-dimensional space, a novel result in geometric graph theory.

Findings

Existence of intersection graphs of axis-aligned boxes with large girth and chromatic number.

Existence of intersection graphs of straight lines with large girth and chromatic number.

These graphs demonstrate complex coloring properties in 3D geometric intersection graphs.

Abstract

We prove that there are intersection graphs of axis-aligned boxes in $\mathbb{R}^3$ and intersection graphs of straight lines in $\mathbb{R}^3$ that have arbitrarily large girth and chromatic number.