On treewidth and maximum cliques
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909883537817600 |
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| author | Chudnovsky, Maria Trotignon, Nicolas |
| author_facet | Chudnovsky, Maria Trotignon, Nicolas |
| contents | 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, 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 $c$, 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]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_07471 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On treewidth and maximum cliques Chudnovsky, Maria Trotignon, Nicolas Combinatorics 05C75, 05C85, 05C69 G.2.2; F.2.2 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, 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 $c$, 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]. |
| title | On treewidth and maximum cliques |
| topic | Combinatorics 05C75, 05C85, 05C69 G.2.2; F.2.2 |
| url | https://arxiv.org/abs/2405.07471 |