A triangulation of a connected closed surface is called weakly regular if the
action of its automorphism group on its vertices is transitive. A triangulation
of a connected closed surface is called degree-regular if each of its vertices
have the same degree. Clearly, a weakly regular triangulation is
degree-regular. In 1999, Lutz has classified all the weakly regular
triangulations on at most 15 vertices. In 2001, Datta and Nilakantan have
classified all the degree-regular triangulations of closed surfaces on at most
11 vertices.
In this article, we have proved that any degree-regular triangulation of the
torus is weakly regular. We have shown that there exists an n-vertex
degree-regular triangulation of the Klein bottle if and only if n is a
composite number ≥9. We have constructed two distinct n-vertex weakly
regular triangulations of the torus for each n≥12 and a (4m+2)-vertex weakly regular triangulation of the Klein bottle for each m≥2. For 12≤n≤15, we have classified all the n-vertex
degree-regular triangulations of the torus and the Klein bottle. There are
exactly 19 such triangulations, 12 of which are triangulations of the torus and
remaining 7 are triangulations of the Klein bottle. Among the last 7, only one
is weakly regular.Comment: Revised version, 26 pages, To appear in Proceedings of Indian Academy
of Sciences (Math. Sci.