Grid Minors and Products

Motivated by recent developments regarding the product structure of planar graphs, we study relationships between treewidth, grid minors, and graph products. We show that the Cartesian product of any two connected $n$-vertex graphs contains an $\mathrm{\Omega }(\sqrt{n})\times \mathrm{\Omega }(\sqrt{n})$ grid minor. This result is tight: The lexicographic product (which includes the Cartesian product as a subgraph) of a star and any $n$-vertex tree has no $\omega (\sqrt{n})\times \omega (\sqrt{n})$ grid minor.