Polynomial treewidth forces a large grid-like-minor

TL;DR

This paper demonstrates that graphs with polynomially large treewidth necessarily contain large grid-like-minors, advancing understanding of the relationship between treewidth and grid minors.

Contribution

It proves that polynomial bounds on treewidth guarantee the existence of large grid-like-minors, improving previous exponential bounds.

Findings

Graphs with treewidth ≥ cℓ^4√logℓ contain a grid-like-minor of order ℓ.

Graphs with large enough treewidth have large grid-like-minors.

Cartesian products G×K_2 contain large K_ℓ minors when G has sufficiently large treewidth.

Abstract

Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an $\ell\times\ell$ grid minor is exponential in $\ell$. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A \emph{grid-like-minor of order} $\ell$ in a graph $G$ is a set of paths in $G$ whose intersection graph is bipartite and contains a $K_{\ell}$-minor. For example, the rows and columns of the $\ell\times\ell$ grid are a grid-like-minor of order $\ell+1$. We prove that polynomial treewidth forces a large grid-like-minor. In particular, every graph with treewidth at least $c\ell^4\sqrt{\log\ell}$ has a grid-like-minor of order $\ell$. As an application of this result, we prove that the cartesian product $G\square K_2$ contains a $K_{\ell}$-minor whenever $G$ has treewidth at least $c\ell^4\sqrt{\log\ell}$.