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    The Z2\mathbb{Z}_2-genus of Kuratowski minors

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    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\mathbb{Z}_2-genus of a graph GG is the minimum gg such that GG has an independently even drawing on the orientable surface of genus gg. An unpublished result by Robertson and Seymour implies that for every tt, every graph of sufficiently large genus contains as a minor a projective t×tt\times t grid or one of the following so-called tt-Kuratowski graphs: K3,tK_{3,t}, or tt copies of K5K_5 or K3,3K_{3,3} sharing at most 22 common vertices. We show that the Z2\mathbb{Z}_2-genus of graphs in these families is unbounded in tt; 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\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 Z2\mathbb{Z}_2-genus of graphs.Comment: 23 pages, 7 figures; a few references added and correcte

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    The genus g(G) of a graph G is the minimum g such that G has an embedding on the orientable surface M_g of genus g. 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 Z_2-genus of a graph G, denoted by g_0(G), is the minimum g such that G has an independently even drawing on M_g. By a result of Battle, Harary, Kodama and Youngs from 1962, the graph genus is additive over 2-connected blocks. In 2013, Schaefer and Stefankovic proved that the Z_2-genus of a graph is additive over 2-connected blocks as well, and asked whether this result can be extended to so-called 2-amalgamations, as an analogue of results by Decker, Glover, Huneke, and Stahl for the genus. We give the following partial answer. If G=G_1 cup G_2, G_1 and G_2 intersect in two vertices u and v, and G-u-v has k connected components (among which we count the edge uv if present), then |g_0(G)-(g_0(G_1)+g_0(G_2))|= m >= 3, we prove that g_0(K_{m,n})/g(K_{m,n})=1-O(1/n). Similar results are proved also for the Euler Z_2-genus. We express the Z_2-genus of a graph using the minimum rank of partial symmetric matrices over Z_2; a problem that might be of independent interest
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