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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, Milanic and Storgel 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 Kc-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.
Chudnovsky et al. (Mon,) studied this question.