Semi-equivelar maps on the torus and the Klein bottle are Archimedean

Datta, Basudeb; Maity, Dipak

doi:10.1016/j.disc.2018.08.016

Cited by 17 publications

(46 citation statements)

References 7 publications

(21 reference statements)

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“…, p n k k ]. A map M is called edge-homogeneous ( [3], we are including the same definition for the sake of completeness) if C u and C v are of same type for all u, v ∈ E(X). More precisely, there exist integers p 1 , .…”

Section: Introduction and Resultsmentioning

confidence: 99%

Semi-equivelar toroidal maps and their k-edge covers

Kundu¹,

Maity²

2021

Preprint

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A map K on a surface is called edge-transitive if the automorphism group of K acts transitively on the set of edges of K. A tiling is edge-homogeneous if any two edges with vertices of congruent face-cycles. In general, edge-homogeneous maps on a surface form a bigger class than edge-transitive maps. There are edge-homogeneous toroidal maps which are not edge-transitive. A map is called minimal if the number of edges is minimal. A map f : M → K is a covering map if f is a covering map from the vertex set of M to the vertex set of K such that f is a surjection and a local isomorphism : the neighbourhood of a vertex v in M is mapped bijectively onto the neighbourhood of f (v) in K. Orbanić et al. and Širáň et al. have shown that every edge-homogeneous toroidal map has edgetransitive cover. In this article, we show the bounds of edge orbits of edge-homogeneous toroidal maps. We also prove that if a edge-homogeneous map is k edge orbital (or k orbital) then it has a finite index m-edge orbital minimal cover for m ≤ k. We also show the existence and classification of n sheeted covers of edge-homogeneous toroidal maps for each n ∈ N.

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Section: Introduction and Resultsmentioning

confidence: 99%

Semi-equivelar toroidal maps and their k-edge covers

Kundu¹,

Maity²

2021

Preprint

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“…, p n k k ], is called vertex type of u. A map M is called semi-equivelar ( [4], we are including the same definition for the sake of completeness) if C u and C v are of same type for all u, v ∈ V (X). More precisely, there exist integers p 1 , .…”

Section: Introductionmentioning

confidence: 99%

“…In fact, there exists [p q ] equivelar maps on R 2 whenever 1/p + 1/q < 1/2 (e.g., [2], [8]). We know from [5,3,4] that the Archimedean tilings E i (1 ≤ i ≤ 11) (in Section 2) are unique as semi-equivelar maps. That is, we have the following.…”

Section: Introductionmentioning

confidence: 99%

“…(e) If X 5 is a m 5 -semiregular toroidal map of type [4 1 , 8 2 ] then there exists a covering η k 5 : Y k 5 → X 5 where Y k 5 is k 5 -semiregular for each k 5 ≤ m 5 such that 3 divides k 5 except (k 5 , m 5 ) = (9, 12), (3,9), (6,9). (f) If X 6 is a m 6 -semiregular toroidal map of type [3 1 , 6 1 , 3 1 , 6 1 ], then there exists a covering (2,10), (4,8), (4, 10), (6,8), (6,10), (10,12), (8,12). (4,20), (8,16), (8,20), (12,16), (12,20), (20, 24), (16, 24).…”

Section: Introductionmentioning

confidence: 99%

“…(f) If X 6 is a m 6 -semiregular toroidal map of type [3 1 , 6 1 , 3 1 , 6 1 ], then there exists a covering (2,10), (4,8), (4, 10), (6,8), (6,10), (10,12), (8,12). (4,20), (8,16), (8,20), (12,16), (12,20), (20, 24), (16, 24). (h) If X 8 is a m 8 -semiregular toroidal map of type [3 2 , 4 1 , 3 1 , 4 1 ], then there exists a covering (15,20), (5,15), (10,15).…”

Section: Introductionmentioning

confidence: 99%

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Semi-equivelar toroidal maps and their k-semiregular covers

Kundu¹,

Maity²

2021

Preprint

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If the face-cycles at all the vertices in a map are of same type then the map is called semi-equivelar. In particular, it is called equivelar if the face-cycles contain same type of faces. A map is semiregular (or almost regular) if it has as few flag orbits as possible for its type. A map is k-regular if it is equivelar and the number of flag orbits of the map k under the automorphism group. In particular, if k = 1, its called regular. A map is k-semiregular if it contains more number of flags as compared to its type with the number of flags orbits k. Drach et al. [7] have proved that every semi-equivelar toroidal map has a finite unique minimal semiregular cover. In this article, we show the bounds of flag orbits of semi-equivelar toroidal maps, i.e., there exists k for each type such that every semi-equivelar map is ℓ-uniform for some ℓ ≤ k. We show that none of the Archimedean types on the torus is semiregular, i.e., for each type, there exists a map whose number of flag orbits is more than its type. We also prove that if a semi-equivelar map is m-semiregular then it has a finite index t-semiregular minimal cover for t ≤ m. We also show the existence and classification of n sheeted k-semiregular maps for some k of semi-equivelar toroidal maps for each n ∈ N.

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