A Hanani-Tutte Theorem for Cycles

TL;DR

This paper proves that if every pair of cycles in a graph has an even crossing number in a drawing, then the graph can be drawn without crossings on a plane or surface, extending Hanani-Tutte type results.

Contribution

It establishes a Hanani-Tutte theorem for cycles, linking even crossing conditions to planarity and surface embeddability, and connects to existing fundamental results.

Findings

Even crossing numbers imply planar drawings.

Extension of Hanani-Tutte theorem to arbitrary surfaces.

Connection to classical surface drawing theorems.

Abstract

Given a drawing $D$ of a graph $G$, we define the crossing number between any two cycles $C_{1}$ and $C_{2}$ in $D$ to be the number of crossings that involve at least one edge from each of $C_1$ and $C_2$ except the crossings between edges that are common to both cycles. We show that if the crossing number between every two cycles in $G$ is even in a drawing of $G$ on the plane, then there is a planar drawing of $G$. This result can be extended to arbitrary surfaces. We also establish an equivalence between our result and a fundamental result due to Cairns-Nikolayevsky and Pelsmajer-Schaefer-\v{S}tefankovi\v{c}, about drawing graphs on surfaces, and derive the Loebl-Masbaum theorem from it.