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

Banff International Research Station for Mathematical Innovation and Discovery

A drawing of a graph on a surface is independently even if every pair of independent 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 An unpublished result by Robertson and Seymour implies that for every $t$, every graph of sufficiently large genus contains as a minor a projective $t\times t$ grid or one of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 $2$ 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 \v{S}tefankovi\v{c}, and giving an approximate version of the Hanani-Tutte theorem on surfaces. Joint work with J. Kyncl.

Mathematics; Convex and discrete geometry; Combinatorics

video/mp4

Author affiliation: IST Austria

2018-08-07

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/66676

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/