Treewidth of the $n \times n$ toroidal grid
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866913148300165120 |
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| author | Gima, Tatsuya Morimoto, Hiraku Okada, Yuto Otachi, Yota |
| author_facet | Gima, Tatsuya Morimoto, Hiraku Okada, Yuto Otachi, Yota |
| contents | In this paper, we show that the treewidth of the $n \times n$ toroidal grid is $2n-1$ for all $n \ge 5$. This closes the gap between the previously known upper bound of $2n-1$ (Ellis and Warren, DAM 2008) and the lower bound of $2n-2$ (Kiyomi, Okamoto, and Otachi, DAM 2016). To establish the matching lower bound, we construct a bramble of maximum order by utilizing maximum components obtained after removing $2n-1$ vertices. Our construction relies on the vertex-isoperimetric properties of the infinite grid to establish tight lower bounds on neighborhood sizes, combined with a careful analysis of balls of radius $n/2-1$ and their boundaries to overcome structural obstructions when $n$ is even. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_21015 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Treewidth of the $n \times n$ toroidal grid Gima, Tatsuya Morimoto, Hiraku Okada, Yuto Otachi, Yota Combinatorics Data Structures and Algorithms In this paper, we show that the treewidth of the $n \times n$ toroidal grid is $2n-1$ for all $n \ge 5$. This closes the gap between the previously known upper bound of $2n-1$ (Ellis and Warren, DAM 2008) and the lower bound of $2n-2$ (Kiyomi, Okamoto, and Otachi, DAM 2016). To establish the matching lower bound, we construct a bramble of maximum order by utilizing maximum components obtained after removing $2n-1$ vertices. Our construction relies on the vertex-isoperimetric properties of the infinite grid to establish tight lower bounds on neighborhood sizes, combined with a careful analysis of balls of radius $n/2-1$ and their boundaries to overcome structural obstructions when $n$ is even. |
| title | Treewidth of the $n \times n$ toroidal grid |
| topic | Combinatorics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2605.21015 |