Burling graphs revisited, part I: New characterizations

TL;DR

This paper introduces five new equivalent characterizations of Burling graphs, a class of triangle-free graphs with high chromatic number, expanding understanding of their geometric and graph-theoretic properties.

Contribution

The paper provides five novel equivalent definitions of Burling graphs, including geometrical, graph-theoretical, and axiomatic characterizations.

Findings

Five new equivalent characterizations of Burling graphs.

Expanded understanding of geometric representations of Burling graphs.

Connections between different definitions of Burling graphs.

Abstract

The Burling sequence is a sequence of triangle-free graphs of increasing chromatic number. Each of them is isomorphic to the intersection graph of a set of axis-parallel boxes in $R^3$. These graphs were also proved to have other geometrical representations: intersection graphs of line segments in the plane, and intersection graphs of frames, where a frame is the boundary of an axis-aligned rectangle in the plane. We call Burling graph every graph that is an induced subgraph of some graph in the Burling sequence. We give five new equivalent ways to define Burling graphs. Three of them are geometrical, one is of a more graph-theoretical flavour and one is more axiomatic.