Nonisomorphic complete triangulations of a surface Original Research Article

This paper is concerned with nonisomorphic triangular embeddings of a complete graph into the same surface. We prove that the minimum order (that is, number of vertices) of such examples is 9 for the nonorientable case, and 12 for the orientable one. We also explore the (nonorientable) case 10, where there are 14 such nonisomorphic triangulations with a remarkable one whose symmetry group is A5. Finally, we exhibit an infinite family of nonisomorphic nonorientable examples.