Degree-regular triangulations of the double-torus

Datta, Basudeb; Upadhyay, Ashish Kumar

doi:10.1515/forum.2006.051

Cited by 14 publications

(333 citation statements)

References 18 publications

(122 reference statements)

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“…Thus a d−equivelar triangulation is a SEM of type (3 d ). A triangulation is called d-covered if each edge of the triangulation is incident with a vertex of degree d. In articles [4] and [5] equivelar triangulations have been studied for Euler characteristics 0 and -2. In [11] Negami and Nakamoto studied d-covered triangulations and asked the question about existence of such triangulations on a surface of Euler charateristic χ with the condition that d = 2 ⌊ 5 + √ 49 − 24χ 2 ⌋ see also [12].…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Example : 7 Some Semi Equivelar Maps on surface of Euler Characteristics -1 : 017, 045, 056, 067, 128, 158, 15u, 236, 267, 278, 34v, 369, 39u, 3uv, 45u, 49u, 49v, 78v, 89v, 0234, 17vu, 5698} 017, 045, 056, 067, 129, 17v, 189, 238, 268, 269, 34v, 389, 39u, 3uv, 45u, 47u, 47v, 568, 5uv, 0234, 185v, 67u9} K 3 = {012, 017, 045, 056, 067, 129, 178, 19v, 238, 268, 269, 34v, 378, 37u, 3uv, 45u, 49u, 49v, 568, 5uv, 0234, 85v1, 67u9} The Graphs EG(G 6 (K 1 )) = ∅, EG(G 2 (K 1 )) = { [2,4], [7,10]}. EG(G 2 (K 2 )) = { [2,4], [3,12]} and EG(G 6 (K 2 )) = { [1,6], [5,7]}. Also, EG(G 2 (K 3 )) = ∅ and EG(G 6 (K 3 )) = { [1,6], [8,12]}.…”

Section: Examplesmentioning

confidence: 99%

“…We see that ψ • φ : T i −→ K i , i = 1, 2, 3 is a covering. Clearly it is a two fold orientable covering : 1,7], [0,6,7], [0,6,13], [0,4,13], [1,2,8], [1,5,8], [1,5,10], [2,7,8], [2,6,7], [2,3,6], [3,6,9], [3,9,10], [3,10,22], [3,4,22], [4,22,15], [4,15,14], [4,14,13], [5,10,19], [5,19,12],…”

Section: Examplesmentioning

confidence: 99%

“…The last column in the tables gives the faces where cylinders are added. 2,3,4], [18,21,19,20]), C 44 ( [1,7,11,10], [16,23,13,12]), C 44 ( [5,6,9,8], [14,15,22,17]) 1,2], [12,13,14]), C 33 ( [ 3,6,9 ], [15,18,21]) C 33 ( [ 4,5,10 ], [16,17,22]), C 33 ( [ 7,8,11 ], [19,20,23]) 2,3,4], [12,14,15,16]), C 44 ( [1,8,5,11],…”

Section: Examplesmentioning

confidence: 99%

“…From here it is also evident that T 1 , T 2 and T 3 are not vertex transitive. Also, since EG(G 4 (N )) = { [1,3], [5,13], [6,8], [9,15], [11,24], [12,23]…”

Section: Proofsmentioning

confidence: 99%

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Semi-equivelar maps

2012

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Semi-Equivelar maps are generalizations of Archimedean Solids (as are equivelar maps of the Platonic solids) to the surfaces other than 2−Sphere. We classify some semi equivelar maps on surface of Euler characteristic −1 and show that none of these are vertex transitive. We establish existence of 12-covered triangulations for this surface. We further construct double cover of these maps to show existence of semi-equivelar maps on the surface of double torus. We also construct several semi-equivelar maps on the surfaces of Euler characteristics −8 and −10 and on non-orientable surface of Euler characteristics −2.

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Section: Introduction and Resultsmentioning

confidence: 99%

Section: Examplesmentioning

confidence: 99%

Section: Examplesmentioning

confidence: 99%

Section: Examplesmentioning

confidence: 99%

“…From here it is also evident that T 1 , T 2 and T 3 are not vertex transitive. Also, since EG(G 4 (N )) = { [1,3], [5,13], [6,8], [9,15], [11,24], [12,23]…”

Section: Proofsmentioning

confidence: 99%

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Semi-equivelar maps

2012

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A New Class of Quantum Codes Associate with a Class of Maps

Bhowmik

Maity

Upadhyay

2022

Proceedings of the Seventh International Conference on Mathematics and Computing

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Degree-regular triangulations of torus and Klein bottle

2005

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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

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Degree-regular triangulations of the double-torus

Cited by 14 publications

References 18 publications

Semi-equivelar maps

Semi-equivelar maps

A New Class of Quantum Codes Associate with a Class of Maps

Degree-regular triangulations of torus and Klein bottle

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