# Degree-regular triangulations of torus and Klein bottle

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## Abstract

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 (4*m* + 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.

## Keywords

Triangulations of 2-manifolds regular simplicial maps combinatorially regular triangulations degree-regular triangulations## Preview

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