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Basudeb Datta, Ashish Kumar Upadhyay

Degree-regular triangulations of the double-torus

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 ℝ³. 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.

Forum Mathematicum, Walter de Gruyter

Print ISSN: 0933-7741
Volume: 18, 11/2006
Pages: 1011 - 1025

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