Treewidth of the $n \times n$ toroidal grid
This resolves an open problem in graph theory concerning the treewidth of toroidal grids, providing a tight bound for a fundamental graph family.
The paper proves that the treewidth of the n×n toroidal grid is exactly 2n-1 for n≥5, closing the gap between the previous upper bound of 2n-1 and lower bound of 2n-2.
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.