Degree-Regular Triangulations of Torus and Klein Bottle-Erratum

[摘要] 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 [8], Lutz has classified all the weakly regular triangulations on at most 15 vertices. In [5], 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.

[学科分类] 数学（综合）

[关键词] Triangulations of 2-manifolds;regular simplicial maps;combinatorially regular triangulations;degree-regular triangulations.