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Degree-regular triangulations of torus and Klein bottle

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 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 nn-vertex degree-regular triangulation of the Klein bottle if and only if nn is a composite number ≥9\geq 9. We have constructed two distinct nn-vertex weakly regular triangulations of the torus for each n≥12n \geq 12 and a (4m+2)(4m + 2)-vertex weakly regular triangulation of the Klein bottle for each m≥2m \geq 2. For 12≤n≤1512 \leq n \leq 15, we have classified all the nn-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.

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