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# Degree-regular triangulations of torus and Klein bottle

Datta, Basudeb and Upadhyay, Ashish Kumar (2005) Degree-regular triangulations of torus and Klein bottle. In: Proceedings of the Indian Academy of Sciences: Mathematical Sciences, 115 (3). pp. 279-307.

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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 [51, Datta and Nilakantan have classified all the degree-regular triangulations of closed surfaces on at most I I vertices.In this article, we have proved that any degree-regular triangulation of the torus is weakly regular. We have shown that there exists ann-vertex degree-regular triangulation of the Klein bottle if and only if n is a composite number \geq 9. We have constructed two distinct n-vertex weakly regular triangulations of the torus for each 11 \geq 12 and a (4m + 2)-vertex weakly regular triangulation of the Klein bottle for each In \geq 2. For 12 \leq n \leq 15, we have classified all then-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.

Item Type: Journal Article Copyright for this article belongs to Indian Academy of Sciences. Triangulations of 2-manifolds;regular simplicial maps;combinatorially regular triangulations;degree-regular triangulations Division of Physical & Mathematical Sciences > Mathematics 28 Sep 2005 19 Sep 2010 04:20 http://eprints.iisc.ernet.in/id/eprint/3729

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