Uniqueness and minimality of large face-width embeddings of graphs

LetG be a graph embedded in a surface of genusg. It is shown that if the face-width of the embedding is at leastclog(g)/loglog(g), then such an embedding is unique up to Whitney equivalence. If the face-width is at leastclog(g), then every embedding ofG which is not Whitney equivalent to our embedding has strictly smaller Euler characteristic.Supported in part by the Ministry of Science and Technology of Slovenia, Research Project P1-0210-101-94.