Embeddings of 3-connected 3-regular planar graphs on surfaces of non-negative Euler characteristic

Whitney's theorem states that every 3-connected planar graph is uniquely embeddable on the sphere. On the other hand, it has many inequivalent embeddings on another surface. We shall characterize structures of a $3$-connected $3$-regular planar graph $G$ embedded on the projective-plane, the torus and the Klein bottle, and give a one-to-one correspondence between inequivalent embeddings of $G$ on each surface and some subgraphs of the dual of $G$ embedded on the sphere. These results enable us to give explicit bounds for the number of inequivalent embeddings of $G$ on each surface, and propose effective algorithms for enumerating and counting these embeddings.Comment: 19 pages, 12 figure

On the genera of polyhedral embeddings of cubic graph

In this article we present theoretical and computational results on the existence of polyhedral embeddings of graphs. The emphasis is on cubic graphs. We also describe an efficient algorithm to compute all polyhedral embeddings of a given cubic graph and constructions for cubic graphs with some special properties of their polyhedral embeddings. Some key results are that even cubic graphs with a polyhedral embedding on the torus can also have polyhedral embeddings in arbitrarily high genus, in fact in a genus {\em close} to the theoretical maximum for that number of vertices, and that there is no bound on the number of genera in which a cubic graph can have a polyhedral embedding. While these results suggest a large variety of polyhedral embeddings, computations for up to 28 vertices suggest that by far most of the cubic graphs do not have a polyhedral embedding in any genus and that the ratio of these graphs is increasing with the number of vertices.Comment: The C-program implementing the algorithm described in this article can be obtained from any of the author

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. © 2007 Elsevier Inc. All rights reserved

On the flexibility of toroidal embeddings

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 \u3c = 16) vertices along one-colored faces and replacing j (j \u3c = 16) other colored faces with planar patches. (c) 2007 Elsevier Inc. All rights reserved

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. © 2007 Elsevier Inc. All rights reserved