Treewidth Forces a Large Grid - like - Minor Bruce

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 G whose 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 graph with treewidth at least cl √ log l has a grid-like-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 cl 4 √ log l.