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The \(\mathbb {Z}_2\)-Genus of Kuratowski Minors

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Abstract

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of a graph G is the minimum g such that G has an independently even drawing on the orientable surface of genus g. An unpublished result by Robertson and Seymour implies that for every t, every graph of sufficiently large genus contains as a minor a projective \(t\times t\) grid or one of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 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 \(\mathbb {Z}_2\)-genus of graphs.

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Notes

  1. Paul Seymour, personal communication (2017)

  2. Paul Seymour, personal communication (2017)

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Acknowledgements

We thank Zdeněk Dvořák, Xavier Goaoc, and Pavel Paták for helpful discussions. We also thank Bojan Mohar, Paul Seymour, Gelasio Salazar, Jim Geelen, and John Maharry for information about their unpublished results related to Conjecture 3.1. Finally we thank the reviewers for corrections and suggestions for improving the presentation.

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  1. Supported by Austrian Science Fund (FWF): M2281-N35. Supported by project 19-04113Y of the Czech Science Foundation (GAČR), by the Czech-French collaboration project EMBEDS II (CZ: 7AMB17FR029, FR: 38087RM), and by Charles University project UNCE/SCI/004.

Authors and Affiliations

  1. IST, Klosterneuburg, Austria

    Radoslav Fulek

  2. Department of Applied Mathematics, Charles University, Faculty of Mathematics and Physics, Malostranské nám. 25, 118 00, Praha 1, Czech Republic

    Jan Kynčl

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  1. Radoslav Fulek
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Correspondence to Jan Kynčl.

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The research was partially performed during the BIRS workshop “Geometric and Structural Graph Theory” (17w5154) in August 2017 and during a workshop on topological combinatorics organized by Arnaud de Mesmay and Xavier Goaoc in September 2017.

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Fulek, R., Kynčl, J. The \(\mathbb {Z}_2\)-Genus of Kuratowski Minors. Discrete Comput Geom 68, 425–447 (2022). https://doi.org/10.1007/s00454-022-00412-w

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  • DOI: https://doi.org/10.1007/s00454-022-00412-w

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