Datta, Basudeb and Upadhyay, Ashish Kumar (2005) Degreeregular triangulations of torus and Klein bottle. In: Proceedings of the Indian Academy of Sciences: Mathematical Sciences, 115 (3). pp. 279307.

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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 degreeregular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degreeregular. In [8], Lutz has classified all the weakly regular triangulations on at most 15 vertices. In [51, Datta and Nilakantan have classified all the degreeregular triangulations of closed surfaces on at most I I vertices.In this article, we have proved that any degreeregular triangulation of the torus is weakly regular. We have shown that there exists annvertex degreeregular triangulation of the Klein bottle if and only if n is a composite number \geq 9. We have constructed two distinct nvertex 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 thenvertex degreeregular 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 

Publication:  Proceedings of the Indian Academy of Sciences: Mathematical Sciences 
Publisher:  Indian Academy of Sciences 
Additional Information:  Copyright for this article belongs to Indian Academy of Sciences. 
Keywords:  Triangulations of 2manifolds;regular simplicial maps;combinatorially regular triangulations;degreeregular triangulations 
Department/Centre:  Division of Physical & Mathematical Sciences > Mathematics 
Date Deposited:  28 Sep 2005 
Last Modified:  19 Sep 2010 04:20 
URI:  http://eprints.iisc.ac.in/id/eprint/3729 
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