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      https://www.ias.ac.in/article/fulltext/pmsc/115/03/0279-0307

    • Keywords

       

      Triangulations of 2-manifolds; regular simplicial maps; combinatorially regular triangulations; degree-regular triangulations

    • 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 ann-vertex degree-regular triangulation of the Klein bottle if and only if n is a composite number ≥ 9. We have constructed two distinctn-vertex weakly regular triangulations of the torus for eachn ≥ 12 and a (4m + 2)-vertex weakly regular triangulation of the Klein bottle for eachm ≥ 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.

    • Author Affiliations

       

      Basudeb Datta1 Ashish Kumar Upadhyay1

      1. Department of Mathematics, Indian Institute of Science, Bangalore - 560 012, India
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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