Excluded Grid Theorem: Improved and Simplified

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+→ Z+, such that for any integer g> 0, any graph of treewidth at least f(g), contains the (g x 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(g98 poly log g) is sufficient to ensure the existence of the (g x g)-grid minor in any graph. In this paper we provide a much simpler proof of the Excluded Grid Theorem, achieving a bound of $f(g)=O(g^{36} poly log g)$. Our proof is self-contained, except for using prior work to reduce the maximum vertex degree of the input graph to a constant.