On the flexibility of toroidal embeddings

Two embeddings Ψ1 and Ψ2 of a graph G in a surface Σ are equivalent if there is a homeomorphism of Σ to itself carrying Ψ1 to Ψ2. 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:(i)Ψ is the unique embedding of G in the torus;(ii)G has three nonequivalent embeddings in the torus, G is the 4-cube Q4 (or C4×C4), and each embedding of G forms a 4-by-4 toroidal grid;(iii)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.