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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.19120 |
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| _version_ | 1866914111021907968 |
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| author | Hajebi, Sepehr |
| author_facet | Hajebi, Sepehr |
| contents | Dallard, Milanič, and Štorgel conjectured that for a hereditary graph class $\mathcal{G}$, if there is some function $f:\mathbb{N}\to\mathbb{N}$ such that every graph $G\in \mathcal{G}$ with clique number $ω(G)$ has treewidth at most $f(ω(G))$, then there is a polynomial function $f$ with the same property. Chudnovsky and Trotignon refuted this conjecture in a strong sense, showing that neither polynomial nor any prescribed growth can be guaranteed in general. Here we prove that, in stark contrast, the analog of the Dallard-Milanič-Štorgel conjecture for pathwidth is true: For every hereditary graph class $\mathcal{G}$, if the pathwidth of every graph in $\mathcal{G}$ is bounded by some function of its clique number, then the pathwidth of every graph in $\mathcal{G}$ is bounded by a polynomial function of its clique number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_19120 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Polynomial bounds for pathwidth Hajebi, Sepehr Combinatorics Dallard, Milanič, and Štorgel conjectured that for a hereditary graph class $\mathcal{G}$, if there is some function $f:\mathbb{N}\to\mathbb{N}$ such that every graph $G\in \mathcal{G}$ with clique number $ω(G)$ has treewidth at most $f(ω(G))$, then there is a polynomial function $f$ with the same property. Chudnovsky and Trotignon refuted this conjecture in a strong sense, showing that neither polynomial nor any prescribed growth can be guaranteed in general. Here we prove that, in stark contrast, the analog of the Dallard-Milanič-Štorgel conjecture for pathwidth is true: For every hereditary graph class $\mathcal{G}$, if the pathwidth of every graph in $\mathcal{G}$ is bounded by some function of its clique number, then the pathwidth of every graph in $\mathcal{G}$ is bounded by a polynomial function of its clique number. |
| title | Polynomial bounds for pathwidth |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.19120 |