Genus of cartesian products of regular bipartite graphs

Let G(n, d) denote a connected regular bipartite graph on 2n vertices and of degree d. It is proved that any Cartesian product G(n, d) × G₁(n₁, d₁) × G₂(n₂, d₂) × ? × G_m(n_m, d_m), such that max {d₁, d₂,…, d_m} ≤ d ≤ d₁ + d₂ + ? + d_m, has a quadrilateral embedding, thereby establishing its genus, and thereby generalizing a result of White. It is also proved that if G is any connected bipartite graph of maximum degree D, if Q_m is the m-cube graph, and if m ≥ D then G × Q_m has a quadrilateral embedding.