Irreducible Triangulations and Triangular Embeddings On a Surface

A classical result of Steinitz says that there is only one irreducible triangulation on the sphere. D. Barnette showed that there are only two irreducible triangulations on the projective plane. Using Barnette’s result, S. Lawrencenko showed that there are at most four different triangular embeddings of a 3-connected graph on the projective plane. In this paper, we give an explicit bound for the number of irreducible triangulations on any surface. We also show that, on any surface, the number of triangular embeddings of a 3connected graph is bounded above by a constant depending only on the surface. Examples show that this does not hold for non-triangular embeddings of 3-connected graphs on the torus.