Combinatorial Structure of Faces in Triangulations on Surfaces

The degree of a vertex or face in a graph on the plane or other orientable surface is the number of incident edges. A face is of type if whenever . We denote the minimum vertex-degree of by . The purpose of our paper is to prove that every triangulation with of the torus, as well as of large enough such a triangulation of any fixed orientable surface of higher genus has a face of one of the types , , , , , or , where all parameters are best possible.