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 An unpublished result by Robertson and Seymour implies that for every $t$, every graph of sufficiently large genus contains as a minor a projective $t\times t$ grid or one of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.

翻译：绘制表面的图表是独立的, 即使绘制中的每对非相近边缘的图纸都具有独立性, 即使每对非相近边缘的图纸在数字交叉点上偶数次数。 美元是G$的美元, 或G$的美元, 这样G$可以独立地在可调整的genus $g美元表面上画一个图。 Robertson和Seymour的未公布结果意味着, 每张足够大的genus的图纸都包含一个微小的投影 $t\time, 或以下所谓的$-Kuratowski的图: $K$3, t$, 或$5美元或K$3, 3美元在大多数两种常见的脊椎上画一个。 我们显示, 这些家族的 $mathbb%2 的图表没有以美元标注; 事实上, 与他们的基因相等。 这表示, 一张图的元与上面的 $$t$- $- kowowski 图表是来自以上的 美元, 或以 equal- equal_b___ ma_ ma_ max max max max max 。