Unified Hanani-Tutte theorem
Radoslav Fulek, Jan Kyn\v{c}l, D\"om\"ot\"or P\'alv\"olgyi

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 has a drawing in the plane where every pair of independent edges crosses an even number of times, then has a planar drawing preserving the rotation of each vertex whose incident edges cross each other evenly in . 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.
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