Criteria for toroidal embedding of one-vertex ribbon graphs

TL;DR

This paper develops criteria for embedding one-vertex ribbon graphs onto a torus, expanding the understanding of graph embeddings on surfaces with a focus on ribbon structures.

Contribution

It provides the first set of criteria specifically for toroidal embedding of one-vertex ribbon graphs, addressing Skopenkov's problem.

Findings

Established necessary and sufficient conditions for toroidal embedding.

Connected ribbon graph structure with surface topology.

Enhanced understanding of graph embeddings on surfaces.

Abstract

The work provides a brief intuitive overview theory of graph on surfaces. We considers graphs with an additional structure, wich we call discs with ribbons, also known as one-vertex ribbon graphs. And solves the problem (Skopenkov's) about criteria for toroidal embedding of one-vertex ribbon graph.