Exponential Families of Non-Isomorphic Triangulations of Complete Graphs

We prove that the number of non-isomorphic face 2-colourable triangulations of the complete graph Kn in an orientable surface is at least 2n2/54−O(n) for n congruent to 7 or 19 modulo 36, and is at least 22n2/81−O(n) for n congruent to 19 or 55 modulo 108.