On treewidth and maximum cliques

TL;DR

This paper constructs specific graph classes to explore relationships between treewidth, clique number, and other properties, providing counterexamples that disprove several existing conjectures in graph theory.

Contribution

It introduces new graph classes that demonstrate bounded treewidth relative to clique number while violating other conjectures, advancing understanding of graph structure complexities.

Findings

Counterexamples to conjectures on treewidth and clique number

Graphs with unbounded tree-independence number despite bounded treewidth

Graphs of large treewidth avoiding high treewidth subgraphs

Abstract

We construct classes of graphs that are variants of the so-called layered wheel. One of their key properties is that while the treewidth is bounded by a function of the clique number, the construction can be adjusted to make the dependance grow arbitrarily. Some of these classes provide counter-examples to several conjectures. In particular, the construction includes hereditary classes of graphs whose treewidth is bounded by a function of the clique number while the tree-independence number is unbounded, thus disproving a conjecture of Dallard, Milani\v{c} and \v{S}torgel [Treewidth versus clique number. II. Tree-independence number. Journal of Combinatorial Theory, Series B, 164:404-442, 2024.]. The construction can be further adjusted to provide, for any fixed integer $c$, graphs of arbitrarily large treewidth that contain no $K_c$-free graphs of high treewidth, thus disproving a conjecture of Hajebi [Chordal graphs, even-hole-free graphs and sparse obstructions to bounded treewidth, arXiv:2401.01299, 2024].