Improved Bounds for the Excluded Grid Theorem

TL;DR

This paper improves the upper bound on the function relating treewidth to the existence of grid minors in graphs, reducing it from a super-exponential to a polynomial bound, using new techniques and a simplified proof.

Contribution

It presents a significantly improved polynomial bound for the Excluded Grid Theorem and introduces new techniques for a simpler, self-contained proof.

Findings

Bound on treewidth function improved to O(g^{19} polylog g)

New techniques enable a simpler proof of the theorem

Achieves polynomial bound on grid minor existence

Abstract

We study the Excluded Grid Theorem of Robertson and Seymour. This is a fundamental result in graph theory, that states that there is some function $f: Z^+\rightarrow Z^+$, such that for all integers $g>0$, every graph of treewidth at least $f(g)$ contains the $(g\times g)$-grid as a minor. Until recently, the best known upper bounds on $f$ were super-exponential in $g$. A recent work of Chekuri and Chuzhoy provided the first polynomial bound, by showing that treewidth $f(g)=O(g^{98}\operatorname{poly}\log g)$ is sufficient to ensure the existence of the $(g\times g)$-grid minor in any graph. In this paper we improve this bound to $f(g)=O(g^{19}\operatorname{poly}\log g)$. We introduce a number of new techniques, including a conceptually simple and almost entirely self-contained proof of the theorem that achieves a polynomial bound on $f(g)$.