On treewidth and maximum cliques

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 dependence 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, Milanič and Štorgel [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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> </mml:math> , graphs of arbitrarily large treewidth that contain no <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> -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].