On the genera of polyhedral embeddings of cubic graph

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