Treewidth of the $n \times n$ toroidal grid

2026-05-20

AI Summary

This study resolves the long-standing open problem concerning the treewidth of the $n \times n$ toroidal grid graph. Prior work established bounds of $2n - 2$ and $2n - 1$, leaving a gap unresolved. By introducing a novel family of brambles and leveraging the vertex isoperimetric inequality on the infinite grid, together with a careful analysis of balls of radius $n/2 - 1$ and their boundaries—augmented by exploiting the graph’s symmetry—we close this gap definitively. Our approach overcomes structural obstacles that arise specifically when $n$ is even, thereby establishing rigorously that for all $n \geq 5$, the treewidth of the toroidal grid is exactly $2n - 1$.

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.

Problem

• treewidth
• toroidal grid
• graph theory
• combinatorial optimization

Innovation

• treewidth
• toroidal grid
• bramble
• vertex-isoperimetric
• graph minor theory