Datta, Basudeb ; Upadhyay, Ashish Kumar (2005) Degreeregular triangulations of torus and klein bottle Proceedings of the Indian Academy of Sciences  Mathematical Sciences, 115 (3). pp. 279307. ISSN 02534142

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Official URL: http://www.ias.ac.in/mathsci/vol115/aug2005/PM2412...
Related URL: http://dx.doi.org/10.1007/BF02829658
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 [5], Datta and Nilakantan have classified all the degreeregular triangulations of closed surfaces on at most 11 vertices. In this article, we have proved that any degreeregular triangulation of the torus is weakly regular. We have shown that there exists an nvertex degreeregular triangulation of the Klein bottle if and only if n is a composite number ≥ 9. We have constructed two distinct nvertex 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 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:  Article 

Source:  Copyright of this article belongs to Indian Academy of Sciences. 
Keywords:  Triangulations of 2manifolds; Regular Simplicial Maps; Combinatorially Regular Triangulations; Degreeregular Triangulations 
ID Code:  22356 
Deposited On:  23 Nov 2010 12:59 
Last Modified:  17 May 2016 06:25 
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