The closed 2-cell embeddings of 2-connected doubly toroidal graphs

AbstractA closed 2-cell embedding of a graph embedded in some surface is an embedding such that each face is bounded by a circuit in the graph. The closed 2-cell embedding conjecture says that every 2-connected graph has a closed 2-cell embedding in some surface. In this paper, we prove that any 2-connected doubly toroidal graph G has a closed 2-cell embedding in some surface. As a corollary, such a graph has a cycle double cover, i.e., G has a set of circuits containing every edge exactly twice