Degree-regular triangulations of torus and Klein bottle

A combinatorial 2-manifold is called weakly regular if the action of its automorphism group on its vertices is transitive. A combinatorial 2-manifold is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular combinatorial 2-manifold is degree-regular. In [5], Lutz has classified all the weakly-regular combinatorial 2-manifolds on at most 15 vertices, among which 20 are of Euler characteristic zero. In [3], Datta and Nilakantan have classified all the degree-regular combinatorial 2-manifolds on at most 11 vertices, among which 8 are of Euler characteristic zero. For 12 ≤ n ≤ 15, we have classified all degree-regular n-vertex combinatorial 2manifolds of Euler characteristic zero. There are exactly 19 such combinatorial 2manifolds, 12 of which triangulate the torus and remaining 7 triangulate the Klein bottle. For each n ≥ 12, we have constructed two distinct weakly regular triangulations of the torus. If n ≥ 9 is not a prime then we have constructed an n-vertex degree-regular triangulation of the Klein bottle. If n is a prime then we have shown that there does not exist any n-vertex weakly regular triangulation of the Klein bottle. AMS classification : 57Q15, 57M20, 57N05.