Burling graphs revisited, part II: Structure

TL;DR

This paper explores the structural properties of Burling graphs, which are triangle-free graphs with high chromatic number, focusing on their geometric representations and implications for graph theory conjectures.

Contribution

It provides new insights into the structure of Burling graphs derived from trees, enhancing understanding of their properties and significance.

Findings

Burling graphs are triangle-free with high chromatic number.

Structural properties of Burling graphs derived from trees are characterized.

Implications for geometric representations and graph theory conjectures are discussed.

Abstract

The Burling sequence is a sequence of triangle-free graphs of increasing chromatic number. Any graph which is an induced subgraph of a graph in this sequence is called a Burling graph. These graphs have attracted some attention because they have geometric representations and because they provide counter-examples to several conjectures about bounding the chromatic number in classes of graphs. We recall an equivalent definition of Burling graphs from the first part of this work: the graphs derived from a tree. We then give several structural properties of derived graphs.