The closed 2-cell embeddings of 2-connected doubly toroidal graphs (Q1901046)

An embedding of a graph \(G\) in a surface \(S\) is called a closed 2-cell embedding if the topological closure of each face of the embedding is homeomorphic to a closed disc. It is proved in this paper that if a 2- connected graph \(G\) can be embedded in a double torus, then \(G\) must have a closed 2-cell embedding in some surface. An immediate corollary of the main theorem is that if a 2-connected graph \(G\) can be embedded in a double torus, then \(G\) has a circuit double cover.