626 research outputs found

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 An unpublished result by Robertson and Seymour implies that for every tt, every graph of sufficiently large genus contains as a minor a projective t×tt\times t grid or one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

    Irreducible triangulations of surfaces with boundary

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    A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly non-orientable) surface of genus g>=0 with b>=0 boundaries is O(g+b). So far, the result was known only for surfaces without boundary (b=0). While our technique yields a worse constant in the O(.) notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boundary
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