Minors and Strong Products

Let G⊠Hand G□H denote, respectively, the strong and Cartesian products of graphs G and H. (We recall thatK2⊠K2 is the complete graph K4on four vertices, whileK2□K2 is a four-cycle C4.) Using a simple construction, we show that, for every bipartite G, the graph G⊠K2is a minor of the graphG□C4 . In particular, the d -cube Qdhas a complete minor on 2⊠(d + 1) / 2vertices if d is odd, and on 3 · 2⊠(d − 2) / 2vertices if d is even. We do not know whether such a complete minor of Qdis largest possible.