Treewidth of the $n \times n$ toroidal grid

TL;DR

This paper proves that the treewidth of the $n imes n$ toroidal grid is exactly $2n-1$ for all $n geq 5$, resolving a long-standing gap between known bounds.

Contribution

It establishes the exact treewidth of the toroidal grid by constructing a maximum order bramble, matching the known upper bound.

Findings

Treewidth of $n imes n$ toroidal grid is $2n-1$ for all $n geq 5$.

Constructed a maximum order bramble using grid properties.

Used vertex-isoperimetric properties to establish tight lower bounds.

Abstract

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.