Doubly Semiequivelar Maps on Torus and Klein Bottle

Abstract

A tiling of the Euclidean plane, by regular polygons, is called 2-uniform tiling if it has two orbits of vertices under the action of its symmetry group. There are 20 distinct 2-uniform tilings of the plane. Plane being the universal cover of torus and Klein bottle, it is natural to ask about the exploration of maps on these two surfaces corresponding to the 2-uniform tilings. We call such maps as doubly semiequivelar maps. In the present study, we compute and classify (up to isomorphism) doubly semiequivelar maps on torus and Klein bottle. This classification of semiequivelar maps is useful in classifying a category of symmetrical maps which have two orbits of vertices, named as 2-uniform maps.

Some doubly semi-equivelar maps on the plane and the torus

“…Suppose, M is a map with two distinct face-sequences f 1 and f 2 . Then M is called a doubly semiequivelar map [11], in short DSEM, if (i) φ(v) has same sign (i.e., either negative, 0 or positive) for all v ∈ V (M ), and (ii) the vertices of same face-sequence also have links of same face-sequence up to a cyclic permutation. We denote the M of type [f (f 11 ,...,f 1r 1 ) 1 : f (f 21 ,...,f 2r 2 ) 2 ], where f 1i , f 2j ∈ {f 1 , f 2 }, for 1 ≤ i ≤ r 1 and 1 ≤ j ≤ r 2 , if vertices of the face-sequence f 1 have links of face-sequence (f 11 , .…”

“…: 3 4 .6] 1 . Their respective DSEM types are given in Table 2 of [11]. For simplicity, the types of these DSEMs are denoted by the same notations as used for the respective tilings.…”

“…Consider a DSEM M of type [3 6 : 3 3 .4 2 ] 2 with the vertex set V (M ). Then the map M exists if [11]), where |V (3 6 ) | and |V (3 3 ,4 2 ) | are same as in Section 3.1. Now define the following three types paths in M as follows.…”

“…| = 4k, for some k ∈ N, see [11]. In this type DSEM, we consider a path of type, say W 1 , as P 1 = P (.…”

Hamiltonicity of doubly semi-equivelar maps on the torus

“…In this article, we deal with the 2-uniform tilings [3 3 .4 4 .6] 1 , [3 6 : 3 4 .6] 2 , [3 3 .4 2 : 3.4.6.4], [3 6 : 3 2 .4.12], [3.4.3.12 : 3.12 2 ], [3.4.6.4 : 4.6.12] and [3.4 2 .6 : 3.4.6.4], which induce infinitely many doubly semi-equivelar maps on torus, see [14,19]. A study of doubly semi-equivelar maps on torus and Klein bottle is also carried out in [15,16].…”