article published yes Radoslav Fulek author 39F3FFE4-F248-11E8-B48F-1D18A9856A870000-0001-8485-1774 Jan Kynčl author UlWa department Eliminating intersections in drawings of graphs project 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 Z2 -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of a graph G is the minimum g such that G has an independently even drawing on the orientable surface of genus g. An unpublished result by Robertson and Seymour implies that for every t, every graph of sufficiently large genus contains as a minor a projective t×t grid or one of the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 two 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. We also obtain an analogous result for Euler genus and Euler Z2-genus of graphs. Springer Nature2022 eng 0179-5376 1432-0444 1803.05085 00082501450000110.1007/s00454-022-00412-w 68425-447 https://research-explorer.ista.ac.at/record/186 Fulek R, Kynčl J. The Z2-Genus of Kuratowski minors. <i>Discrete and Computational Geometry</i>. 2022;68:425-447. doi:<a href="https://doi.org/10.1007/s00454-022-00412-w">10.1007/s00454-022-00412-w</a> R. Fulek, J. Kynčl, Discrete and Computational Geometry 68 (2022) 425–447. R. Fulek and J. Kynčl, “The Z2-Genus of Kuratowski minors,” <i>Discrete and Computational Geometry</i>, vol. 68. Springer Nature, pp. 425–447, 2022. Fulek, Radoslav, and Jan Kynčl. “The Z2-Genus of Kuratowski Minors.” <i>Discrete and Computational Geometry</i>, vol. 68, Springer Nature, 2022, pp. 425–47, doi:<a href="https://doi.org/10.1007/s00454-022-00412-w">10.1007/s00454-022-00412-w</a>. Fulek, Radoslav, and Jan Kynčl. “The Z2-Genus of Kuratowski Minors.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2022. <a href="https://doi.org/10.1007/s00454-022-00412-w">https://doi.org/10.1007/s00454-022-00412-w</a>. Fulek, R., &#38; Kynčl, J. (2022). The Z2-Genus of Kuratowski minors. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href="https://doi.org/10.1007/s00454-022-00412-w">https://doi.org/10.1007/s00454-022-00412-w</a> Fulek R, Kynčl J. 2022. The Z2-Genus of Kuratowski minors. Discrete and Computational Geometry. 68, 425–447. 115932022-07-17T22:01:56Z2025-04-14T13:52:37Z