On the flexibility of toroidal embeddings

Two embeddings Ψ 1 and Ψ 2 of a graph G in a surface Σ are equivalent if there is a homeomorphism of Σ to itself carrying Ψ 1 to Ψ 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:(i) he unique embedding of G in the torus; three nonequivalent embeddings in the torus, G is the 4-cube Q 4 (or C 4 × C 4 ), and each embedding of G forms a 4-by-4 toroidal grid; 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.