Degree-regular triangulations of the double-torus

Datta, B and Upadhyay, AK (2006) Degree-regular triangulations of the double-torus. In: Forum Mathematicum, 18 (6). pp. 1011-1025.

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Abstract

A connected combinatorial 2-manifold is called degree-regular if each of its vertices have the same degree. A connected combinatorial 2-manifold is called weakly regular if it has a vertex-transitive automorphism group. Clearly, a weakly regular combinatorial 2-manifold is degree-regular and a degree-regular combinatorial 2-manifold of Euler characteristic -2 must contain 12 vertices. In 1982, McMullen et al. constructed a 12-vertex geometrically realized triangulation of the double-torus in R-3. As an abstract simplicial complex, this triangulation is a weakly regular combinatorial 2-manifold. In 1999, Lutz showed that there are exactly three weakly regular orientable combinatorial 2-manifolds of Euler characteristic -2. In this article, we classify all the orientable degree-regular combinatorial 2-manifolds of Euler characteristic -2. There are exactly six such combinatorial 2-manifolds. This classifies all the orientable equivelar polyhedral maps of Euler characteristic -2.

Item Type:	Journal Article
Additional Information:	Copyright of this article belongs to Walter de Gruyter GmbH & Co. KG.
Keywords:	Triangulations of 2-manifolds;regular simplicial maps; combinatorially regular triangulations;degree-regular triangulations.
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	07 Aug 2007
Last Modified:	24 Jan 2012 05:36
URI:	http://eprints.iisc.ernet.in/id/eprint/11416

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