2004/03/25 by Basudeb Datta, A. Upadhyay, Datta, Basudeb +2
Computer Science · Materials Science · Mathematics · #57M20 #57N05 #57Q15 #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Supramolecular Self-Assembly in Materials #Topological and Geometric Data Analysis #math.AT #math.GT #msc:57M20 #msc:57N05 #msc:57Q15
paper · pdf · doi:10.48550/arxiv.math/0403433
Revised version, 26 pages, To appear in Proceedings of Indian Academy of Sciences (Math. Sci.)
openalex publication_date 2004/03/25 · arxiv created 2005/07/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 1999, Lutz has classified all the weakly regular triangulations on at most 15 vertices. In 2001, 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 the n-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.