# Burling graphs, chromatic number, and orthogonal tree-decompositions

## TL;DR

This paper demonstrates that for certain graphs, having two tree-decompositions with small intersections does not bound the chromatic number, contrasting with known results for path-decompositions.

## Contribution

It provides a counterexample showing that two tree-decompositions with small intersections do not guarantee bounded chromatic number, extending Burling's triangle-free graphs.

## Key findings

- Graphs with large chromatic number can have two tree-decompositions with intersections of at most two vertices.
- This remains true even if one decomposition is a path-decomposition.
- Counterexamples are constructed using Burling's triangle-free graphs.

## Abstract

A classic result of Asplund and Gr\"unbaum states that intersection graphs of axis-aligned rectangles in the plane are $\chi$-bounded. This theorem can be equivalently stated in terms of path-decompositions as follows: There exists a function $f:\mathbb{N}\to\mathbb{N}$ such that every graph that has two path-decompositions such that each bag of the first decomposition intersects each bag of the second in at most $k$ vertices has chromatic number at most $f(k)$. Recently, Dujmovi\'c, Joret, Morin, Norin, and Wood asked whether this remains true more generally for two tree-decompositions. In this note we provide a negative answer: There are graphs with arbitrarily large chromatic number for which one can find two tree-decompositions such that each bag of the first decomposition intersects each bag of the second in at most two vertices. Furthermore, this remains true even if one of the two decompositions is restricted to be a path-decomposition. This is shown using a construction of triangle-free graphs with unbounded chromatic number due to Burling, which we believe should be more widely known.