Degree-regular triangulations of torus and Klein bottle

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 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.Proposition 1 . There are exactly 77 weakly regular combinatorial 2-manifolds on at most 15 vertices; 42 of these are orientable and 35 are non-orientable. Among these 77 combinatorial 2-manifolds, 20 are of Euler characteristic 0. These 20 are T

Degree-regular triangulations of the double-torus

“…Subcase 1.1. In the first case, considering lk(8), we get (z, w) = (9,8). So, lk(4) = C 7 (u, 3, 0, 5, v, 9, 8).…”

“…We denote this by (0, 1) (3,8) (4, u, 5, 9, 6) : (9, u, 6) ∼ = (4, 9, u). With this notation we have : (0, 1) (3,8) (4, u, 5) (6,9) : (9, u, 5) ∼ = (9, 4, u), (0, 1)(3, 8)(4, u) (5,9,6) : (9,6, u) ∼ = (5, 9, u), (0, 1)(2, 7)(3, 9) (6,8) (4, 5, u) : (9, u, 4) ∼ = (9, 5, u), (0, 1) (3, 8)(9, 6, 5) : (9,6,4) ∼ = (5, 9, 4), (0, 1)(2, 7) (3, 5, u, 4, 9) (6,8) : (9, u, 3) ∼ = (5, 9, u), (0, 1) (3, 8)(4, 9, 6, 5) : (9,6,5) ∼ = (5,4,9), considering the links of u, 4 and 5, we see that 458 or 45v is a face. Thus, lk(4) = (u, 3, 0, 5, v, z, w) or C 7 (u, 3, 0, 5, 8, z, w) for some z, w ∈ V .…”

“…There exists an n-vertex degree-regular triangulation of the Klein bottle if and only if n is a composite number ≥ 9 and for k ≥ 2, there exists a (4k + 2)-vertex weakly regular triangulation of the Klein bottle (cf. [8,9]). There are infinitely many combinatorially regular triangulations of the torus whereas there is no such triangulation of the Klein bottle (cf.…”

There is no triangulation of the torus with vertex degrees 5, 6, ... , 6, 7 and related results: geometric proofs for combinatorial theorems

“…This observation allows us to give an easy proof of the classification of (degree-)regular tilings of the torus. This has been treated by many authors [2,16,17,5,4], although most of the results are restricted to the class of polyhedral maps.…”

“…Proof (of Theorems [1][2][3][4][5] In the situations of Theorems 1-5, the holonomy group is a subgroup of Cn with n := 6, 4, 3, 3, 2, respectively, according to Lemmas 10 and 11. On the other hand, the assumptions on the exceptional vertices imply that the equilateral metric is a euclidean cone metric with two cone points of curvature ±2π/n.…”