Burling graphs revisited, part III: Applications to $\chi$-boundedness

TL;DR

This paper explores Burling graphs and their applications to $$-boundedness, identifying specific graphs that are not weakly pervasive, thereby advancing understanding of graph coloring properties.

Contribution

It applies structural results of Burling graphs to identify new non-Burling graphs and their implications for $$-boundedness.

Findings

$K_5$ is not weakly pervasive

Certain series-parallel graphs called necklaces are not weakly pervasive

The results connect Burling graphs with $$-boundedness theory

Abstract

The Burling sequence is a sequence of triangle-free graphs of unbounded chromatic number. The class of Burling graphs consists of all the induced subgraphs of the graphs of this sequence. In the first and second parts of this work, we introduced derived graphs, a class of graphs, equal to the class of Burling graphs, and proved several geometric and structural results about them. In this third part, we use those results to find some Burling and non-Burling graphs, and we see some applications of this in the theory of $\chi$-boundedness. In particular, we show that several graphs, like $K_5$, some series-parallel graphs that we call necklaces, and some other graphs are not weakly pervasive.