Abstrakt
An unpublished result by Robertson and Seymour implies that for every t, every graph of sufficiently large genus contains as a minor a projective t x t grid or one of the following so-called t-Kuratowski graphs: K3,t, or t copies of K5 or K3,3 sharing at most 2 common vertices. We show that the Z2-genus of graphs in these families is unbounded in t; in fact, equal to their genus.
Together, this implies that the genus of a graph is bounded from above by a function of its Z2-genus, solving a problem posed by Schaefer and Štefankovič, and giving an approximate version of the Hanani-Tutte theorem on orientable surfaces.