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 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.