Polynomial treewidth forces a large grid-like-minor

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 l × l grid minor is exponential in l. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A grid-like-minor of order l in a graph G is a set of paths in Gwhose intersection graph is bipartite and contains a Kl-minor. For example, the rows and columns of the l × l grid are a grid-like-minor of order l + 1. We prove that polynomial treewidth forces a large grid-like-minor. In particular, every graphwith treewidth at least cl4 √ log lhas a gridlike-minor of order l. As an application of this result, we prove that the Cartesian product G K2 contains a Kl-minor whenever G has treewidth at least cl4 √ log l. © 2011 Published by Elsevier Ltd