  A drawing of a graph on a surface is independently even if every pair of
nonadjacent edges in the drawing crosses an even number of times. The
$\mathbb{Z}_2$-genus of a graph $G$ is the minimum $g$ such that $G$ has an
independently even drawing on the orientable surface of genus $g$. An
unpublished result by Robertson and Seymour implies that for every $t$, every
graph of sufficiently large genus contains as a minor a projective $t\times t$
grid or one of the following so-called $t$-Kuratowski graphs: $K_{3,t}$, or $t$
copies of $K_5$ or $K_{3,3}$ sharing at most two common vertices. We show that
the $\mathbb{Z}_2$-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 $\mathbb{Z}_2$-genus, solving a problem
posed by Schaefer and \v{S}tefankovi\v{c}, and giving an approximate version of
the Hanani-Tutte theorem on orientable surfaces. We also obtain an analogous
result for Euler genus and Euler $\mathbb{Z}_2$-genus of graphs.

Combinatorics;Discrete Mathematics