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.

翻译：本文证明：对所有 $n \ge 5$，$n \times n$ 环形网格的树宽为 $2n-1$。这一结果填补了此前已知上界 $2n-1$（Ellis 与 Warren，DAM 2008）与下界 $2n-2$（Kiyomi、Okamoto 与 Otachi，DAM 2016）之间的空白。为建立匹配的下界，我们通过利用移除 $2n-1$ 个顶点后获得的最大连通分支构造了一个最大阶的荆棘。该构造依赖于无限网格的顶点等周性质来建立邻域规模的紧下界，并通过对半径为 $n/2-1$ 的球及其边界进行细致分析，克服了 $n$ 为偶数时的结构障碍。