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arXiv · 2405.19274

A Hanani-Tutte Theorem for Cycles

Sutanoya Chakraborty·Arijit Ghosh
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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.

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Sutanoya Chakraborty, Arijit Ghosh. 2024-05-29. A Hanani-Tutte Theorem for Cycles. https://arxiv.org/abs/2405.19274

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