Counterexample to an extension of the Hanani-Tutte theorem on the surface of genus 4

Radoslav Fulek, Jan Kynčl

Published 2017-09-01Version 1

We find a graph of genus $5$ and its drawing on the orientable surface of genus $4$ with every pair of independent edges crossing an even number of times. This shows that the strong Hanani-Tutte theorem cannot be extended to the orientable surface of genus $4$. As a base step in the construction we use a counterexample to the unified Hanani-Tutte theorem on the torus.