Degree-regular triangulations of torus and klein bottle

Datta, Basudeb ; Upadhyay, Ashish Kumar (2005) Degree-regular triangulations of torus and klein bottle Proceedings of the Indian Academy of Sciences - Mathematical Sciences, 115 (3). pp. 279-307. ISSN 0253-4142

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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 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 (4m + 2)-vertex weakly regular triangulation of the Klein bottle for each m ≥ 2. For 12 ≤ n ≤ 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:Article
Source:Copyright of this article belongs to Indian Academy of Sciences.
Keywords:Triangulations of 2-manifolds; Regular Simplicial Maps; Combinatorially Regular Triangulations; Degree-regular Triangulations
ID Code:22356
Deposited On:23 Nov 2010 12:59
Last Modified:17 May 2016 06:25

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