Tree-alpha and excluding finitely many graphs
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910185004466176 |
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| author | Hajebi, Sepehr Spirkl, Sophie |
| author_facet | Hajebi, Sepehr Spirkl, Sophie |
| contents | We prove that a hereditary graph class $\mathcal{G}$ defined by finitely many excluded induced subgraphs has bounded tree-$α$ if and only if it is "$(\mathrm{tw},ω)$-bounded" (that is, for all $t\in \mathbb N$, the class of all $K_t$-free graphs in $\mathcal{G}$ has bounded treewidth). Equivalently, $\mathcal{G}$ has bounded tree-$α$ if and only if it excludes a complete bipartite graph, a forest whose components each have at most three leaves, and the line graph of such a forest.
This resolves two conjectures of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht: the above, and a weaker one that for all $a,b\in \mathbb N$, every hereditary class that excludes $K_{a,a}$ and the $b$-vertex path has bounded tree-$α$. The latter was already open even for $(a,b)\in \{(2,7),(3,5)\}$, and only recently proved for $(a,b)=(2,6)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_01223 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Tree-alpha and excluding finitely many graphs Hajebi, Sepehr Spirkl, Sophie Combinatorics We prove that a hereditary graph class $\mathcal{G}$ defined by finitely many excluded induced subgraphs has bounded tree-$α$ if and only if it is "$(\mathrm{tw},ω)$-bounded" (that is, for all $t\in \mathbb N$, the class of all $K_t$-free graphs in $\mathcal{G}$ has bounded treewidth). Equivalently, $\mathcal{G}$ has bounded tree-$α$ if and only if it excludes a complete bipartite graph, a forest whose components each have at most three leaves, and the line graph of such a forest. This resolves two conjectures of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht: the above, and a weaker one that for all $a,b\in \mathbb N$, every hereditary class that excludes $K_{a,a}$ and the $b$-vertex path has bounded tree-$α$. The latter was already open even for $(a,b)\in \{(2,7),(3,5)\}$, and only recently proved for $(a,b)=(2,6)$. |
| title | Tree-alpha and excluding finitely many graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.01223 |