On the flexibility of toroidal embeddings

Abstract

Two embeddings ${\Psi }_1$ and ${\Psi }_2$ of a graph G in a surface Σ are equivalent if there is a homeomorphism of Σ to itself carrying ${\Psi }_1$ to ${\Psi }_2$. In this paper, we classify the flexibility of embeddings in the torus with representativity at least 4. We show that if a 3-connected graph G has an embedding Ψ in the torus with representativity at least 4, then one of the following holds:

• Ψ is the unique embedding of G in the torus;

• G has three nonequivalent embeddings in the torus, G is the 4-cube $Q_4$ (or $C_4\times C_4$), and each embedding of G forms a 4-by-4 toroidal grid;

• G has two nonequivalent embeddings in the torus, and G can be obtained from a toroidal 4-by-4 grid (faces are 2-colored) by splitting i $(i⩽16)$ vertices along one-colored faces and replacing j $(j⩽16)$ other colored faces with planar patches.