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

TL;DR

This paper investigates the relationship between the $bZ_2$-genus and the genus of graphs, showing they are closely related for certain minors, which advances understanding of graph embeddings and solves a problem posed by Schaefer and Štefankovič.

Contribution

The paper proves that the genus of a graph can be bounded by its $bZ_2$-genus, providing an approximate Hanani-Tutte theorem for orientable surfaces and addressing an open problem.

Findings

The $bZ_2$-genus of certain Kuratowski minors is unbounded and equals their genus.

The genus of a graph is bounded above by a function of its $bZ_2$-genus.

Results extend to Euler genus and Euler $bZ_2$-genus.

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 An unpublished result by Robertson and Seymour implies that for every $t$, every graph of sufficiently large genus contains as a minor a projective $t\times t$ grid or one of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 \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 $\mathbb{Z}_2$-genus of graphs.