One of the problems in topological graph theory is to find the genus of a given graph. That is, the smallest genus among all genera of surfaces on which the graph can be embedded. There is no general formula which gives the genus, but partial answers are given for special families of graphs.;In this work, we use one of the techniques employed to embed graphs in oriented surfaces, namely the theory of wrapped coverings and excess-current graphs, to find new genus embeddings for a large class of bipartite graphs. Then we generalize the duality theorem of wrapped coverings and the theory of wrapped coverings and excess-current graphs to embed certain families of graphs in nonorientable surfaces. This generalized theory enables us to find new nonorientable genus embeddings of many families of graphs including some bipartite graphs.
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