The Z2-Genus of Kuratowski minors
Fulek, Radoslav
Kynčl, Jan
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of a graph G is the minimum g such that G has an independently even drawing on the orientable surface of genus g. An unpublished result by Robertson and Seymour implies that for every t, every graph of sufficiently large genus contains as a minor a projective t×t grid or one of the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 Nature
2022
info:eu-repo/semantics/article
doc-type:article
text
http://purl.org/coar/resource_type/c_6501
https://research-explorer.ista.ac.at/record/11593
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>
eng
info:eu-repo/semantics/altIdentifier/doi/10.1007/s00454-022-00412-w
info:eu-repo/semantics/altIdentifier/issn/0179-5376
info:eu-repo/semantics/altIdentifier/issn/1432-0444
info:eu-repo/semantics/altIdentifier/wos/000825014500001
info:eu-repo/semantics/altIdentifier/arxiv/1803.05085
info:eu-repo/grantAgreement/FWF//M02281
info:eu-repo/semantics/openAccess