Unified Hanani-Tutte theorem

TL;DR

This paper presents a unified generalization of the Hanani-Tutte theorems, providing a new, simpler proof that relates even crossings in a graph drawing to the existence of a planar drawing preserving vertex rotations.

Contribution

It introduces a unified theorem combining strong and weak Hanani-Tutte results and offers a new, simpler proof of this generalization.

Findings

The theorem generalizes existing Hanani-Tutte results.

A new proof simplifies understanding of the theorem.

The result connects even crossings to planar embeddings.

Abstract

We introduce a common generalization of the strong Hanani-Tutte theorem and the weak Hanani-Tutte theorem: if a graph $G$ has a drawing $D$ in the plane where every pair of independent edges crosses an even number of times, then $G$ has a planar drawing preserving the rotation of each vertex whose incident edges cross each other evenly in $D$. The theorem is implicit in the proof of the strong Hanani-Tutte theorem by Pelsmajer, Schaefer and \v{S}tefankovi\v{c}. We give a new, somewhat simpler proof.