Minors and Strong Products

Let G H and G2H denote, respectively, the strong and Cartesian products of graphs G and H . (We recall that K2 K2 is the complete graph K4 on four vertices, while K22K2 is a four-cycle C4.) Using a simple construction, we show that, for every bipartite G, the graph G K2 is a minor of the graph G2C4. In particular, the d-cube Qd has a complete minor on 2 (d+1)/2 vertices if d is odd, and on 3 · 2(d−2)/2 vertices if d is even. We do not know whether such a complete minor of Qd is largest possible.