On the flexibility of toroidal embeddings
Abbreviated Journal Title
J. Comb. Theory Ser. B
embedding; torus; flexibility; representativity; GRAPHS; UNIQUENESS; WIDTH; Mathematics
Two embeddings psi(1) and psi(2) of a graph G in a surface Sigma are equivalent if there is a homeomorphism of Sigma to itself carrying psi(1) to psi(2). In this paper, we classify the flexibility of ernbeddings in the torus with representativity at least 4. We show that if a 3-connected graph G has an embedding psi in the torus with representativity at least 4, then one of the following holds: (i) psi 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 x 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. (c) 2007 Elsevier Inc. All rights reserved.
Journal of Combinatorial Theory Series B
"On the flexibility of toroidal embeddings" (2008). Faculty Bibliography 2000s. 898.