Burling graphs revisited, part II: Structure

TL;DR

The paper advances the structural theory of Burling graphs by exploiting the derived-graph viewpoint to obtain a decomposition theorem for oriented Burling graphs via star cutsets, and by characterizing subdivisions of $K_4$ that remain Burling. It also introduces and connects two equivalent formulations of Burling graphs: $k$-Burling graphs (via nobility) and $k$-sequential graphs (via base forests and top-sets), establishing a robust framework for analyzing triangle-free graphs with unbounded chromatic number. The work yields new non-Burling graph classes (including wheels and the Theta+ graph) and gives a precise subdivision criterion for $K_4$, with consequences for $\chi$-boundedness questions in wheel-free graphs. The results have implications for the broader study of weakly pervasive graphs and set the stage for applications in the paper’s Part III.

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.

Figures (18)

• Figure 1: Complete bipartite graphs derived from trees.
• Figure 2: Cycle of length 6 derived from a tree.
• Figure 3: Transforming $v$ into a non-last-born.
• Figure 4: Transforming $v$ into a last-born.
• Figure 5: Subdividing a bottom-arc.

Theorems & Definitions (61)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5
• proof