Uniform strong 2-cell embeddings of bridgeless graphs (Q1897438)

By a strong 2-cell embedding of a graph \(G\) is meant an open 2-cell embedding of \(G\) in which the image of every edge of \(G\) is incident with exact two faces of the embedding. An embedding is said to be uniform if the boundary of each face contains the same number of edges. If the number is specified to be \(k\) in a uniform embedding, then the embedding is called a \(k\)-pattern. In this paper, the authors illustrate the existence of \(k\)-patterns for all values of \(k\geq 3\) and all possible numbers of faces on the sphere, the projective plane, the torus and the Klein bottle. Then, they also show some sharp characterizations of uniform embeddings as \(k\)-patterns based on the number of edges on each face boundary.