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.

1. Introduction

Equivelar and semiequivelar maps are generalizations of the maps on the surfaces of well-known Platonic solids and Archimedean solids to the closed surfaces other than the 2-sphere, respectively. A substantial literature is available for such maps (see [1–8]).

Tilings of the plane are a great source of polyhedral maps on the surfaces of torus and Klein bottle, as the plane is the universal cover of these two surfaces. A tiling of the plane, by regular polygons, is called a k-uniform tiling if it has k orbits of vertices under its symmetry. The k-uniform tilings have been completely enumerated for k ≤ 6. There are 11 1-uniform, 20 2-uniform, 61 3-uniform, 151 4-uniform, 332 5-uniform, and 673 6-uniform tilings on the plane. For a detailed study on such tilings, readers are referred to see [9–11].

The 11 1-uniform tilings of the plane are also called Archimedean tilings. Out of these, 3 are regular and 8 are semiregular tilings. The 3 regular tilings provide equivelar maps of types [3⁶], [4⁴], and [6³] and 8 semiregular tilings provide semiequivelar maps of types [3⁴, 6], [3³, 4²], [3², 4, 3, 4], [3, 4, 6, 4], [3, 6, 3, 6], [3, 12²], [4, 6, 12], and [4, 8²] on torus and Klein bottle. Altshuler [12] has given a construction for a map of the type [3⁶] and [6³] on the torus. Kurth [13] has enumerated maps of the types [3⁶], [4⁴], and [6³] on the torus. In [2], Datta and Nilakantan classified map of type [3⁶] and [4⁴] on at most 11 vertices. In continuation of this, Datta and Upadhyay [14] classified these type of maps for n vertices with 12 ≤ n ≤ 15. In [15], Brehm and Kuhnel have classified these three types equivelar maps on the torus using a different approach. In [16], Tiwari and Upadhyay have classified the 8 types semiequivelar maps on at most 20 vertices. Recently, Maity and Upadhyay [17] have presented a way to classify the eight types of semiequivelar maps on the torus for arbitrary number of vertices.

Analogues to the Archimedean tilings, here we initiate the theory of maps on torus and Klein bottle corresponding to the 2-uniform tilings. We call such maps as doubly semiequivelar map(s) or briefly DSEM(s). The present work provides a new class of polyhedra which have two classes of vertices in terms of the arrangement of polygons around the vertices. Polyhedra play an important role in human life. It has extensive application in ornament designing, architectural designing, cartography, computer graphics etc. (see [18–20]).

This article is organized as follows: In Section 2, we give basic definitions and notations used in the present work. In Section 3, we define doubly semiequivelar map (DSEM) and describe a methodology to enumerate a doubly semiequivelar map on torus and Klein bottle. In Section 4, we compute and classify DSEMs on torus and Klein bottle. In Section 5, we present the results obtained from the computation and classification. A tabular form of the results is shown in Table 1. In Section 6, we present discussion and future scope of the DSEMs followed by some concluding remarks.

Table 1 

DSEMs of face-size 4 on torus and Klein bottle on ≤15 vertices.

2. Basic Definitions and Notations

For graph theory related terminologies, we refer [21]. A p-cycle, denoted as C_p, is a 2-regular graph with p vertices. We denote C_p explicitly as C_p(), where the vertex set V(C_p) = {} and edge set E(C_p) = {}.

A surface (closed surface) F is a connected, compact 2-manifold without boundary. A surface F is either sphere, sphere with handles (also called orientable surface of genus , denoted as ), or sphere with cross caps (also called nonorientable surface of genus , denoted as ). To a surface, we associate a unique integer called its Euler characteristic χ and is defined as χ() = 2 −  and χ() = 2 − . The surfaces S₁ and N₂ of Euler characteristic 0 are called torus and Klein bottle, respectively.

An embedding of a connected, simple graph into a surface F is called 2-cell embedding if the closure of each connected component of F\G is a 2-disk D_p. These components are called faces of the embedding. The vertices and edges of G are called the vertices and edges of the embedding. A map (polyhedral) M on a surface F is a 2-cell embedding such that the nonempty intersection of any two faces is either a vertex or an edge [22]. The face size of a map M is p, if p is the largest positive integer such that M has a face D_p.

Two maps M₁ and M₂, with vertex sets V(M₁) and V(M₂), respectively, are said to be isomorphic if there is a bijective map f: V(M₁) ⟶ V(M₂) which preserves the incidence of edges and incidence of faces. An isomorphism from a map M to itself is also called an automorphism. A collection Aut(M) of all the automorphisms of a map M forms a group under the composition of maps, called the automorphism group of M. A map M is called vertex-transitive if it has a unique orbit of vertices under the action of Aut(M).

The face-sequence [7] of a vertex , denoted as , in a map M is a finite cyclic sequence , where p₁, …, p_k ≥ 3 and n₁, …, n_k ≥ 1, such that the face cycle at is . A map is called semiequivelar of type if the face-sequence of each vertex is . A semiequivelar map of type [pⁿ] is also called equivelar map.

Let be the face cycle at a vertex in a map M. Let denote the boundary cycles of these . Then, the link of , denoted as , is a cycle in M consisting of all the vertices of these ’s except and all the edges of these ’s except which has one end vertex . If is a vertex with , the face-sequence of is a cyclically ordered sequence (f − seq(), …, f − seq()).

Let be a vertex with the face-sequence . The combinatorial curvature of , denoted by , is defined as .

3. Definition of the Problem and Description of Method

Let M be a map with two distinct face-sequences f₁ and f₂. We say that M is a doubly semiequivelar map, in short DSEM, if (i) the sign of ϕ() is same for all  ∈ M and (ii) vertices of same type face-sequence also have links of the same face-sequence up to a cyclic permutation. A doubly semiequivelar map M is called 2-uniform if it has 2 orbits of vertices under the action of its automorphism group. We denote the M of type , where f_1i or f_2j is f₁ or f₂, for 1 ≤ i ≤ r₁ and 1 ≤ j ≤ r₂, if vertices of the face-sequence f₁ have links of face-sequence and vertices of the face-sequence f₂ have links of face-sequence , respectively.

There are 20 types of 2-uniform tilings of the plane denoted as , , [3⁶: 3², 4, 3, 4], , , , , , , [3⁶: 3², 4, 12], [3⁶: 3², 6²], [3⁴, 6: 3², 6²], , [3², 4, 3, 4, 4²: 3, 4, 6, 4], [3², 6²: 3, 6, 3, 6], [3, 4, 3, 12 : 3, 12²], [3, 4², 6 : 3, 4, 6, 4], , , [3, 4, 6, 4 : 4, 6, 12] (see [11]). Out of these, the first seven types have p-gons, with p ≤ 4 (see Figure 1).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Figure 1 

2-uniform tilings of types: , , , , , , .

3.1. 2-Uniform Tilings of the Plane

We classify the DSEMs on torus and Klein bottle corresponding to the above seven tilings. We abbreviate the types of DSEMs by the same notation as used for the respective tilings (see Table 2).

Table 2 

Tabulated list of DSEMs of face-size 4.

3.2. Methodology

Each doubly semiequivelar map, out of the seven types (listed in Table 2), contains two types of face-sequences among the four types (3⁶), (3³, 4²), (3², 4, 3, 4), and (4⁴) around the vertices. We use the following notations frequently to denote a vertex with specific type face-sequence in the computation. Here lk() means link of vertex .(i)The notation lk() = C₆(a, b, c, d, e, f) means the face-sequence of is (3⁶), i.e., the triangular faces [, b, c], [, c, d], [, d, e], [, e, f], [, f, a], and [, b, a] are incident at .(ii)lk() = C₇(a, b, [c, d, e, f, ]) means the face-sequence of is (3³, 4²), i.e., the triangular faces [, a, ], [, a, b], and [, b, c] and quadrangular faces [, c, d, e] and [, e, f, ] are incident at .(iii)lk() = C₇(a, [b, c, d], [e, f, ]) means the face-sequence of is (3², 4, 3, 4), i.e., the triangular faces [, a, b], [, a, ], and [, d, e] and quadrangular faces [, b, c, d] and [, e, f, ] are incident at .(iv)lk() = C₈(a, b, c, d, e, f, , h) means the face-sequence of is (4⁴), i.e., the quadrangular faces [, a, b, c], [, c, d, e], [, e, f, ], and [, , h, a] are incident at .

Since a doubly semiequivelar map contains two types of vertices, in terms of face-sequences, to distinguish these vertices, we denote vertices of one type face-sequence by n and the other type by a_n, for some . We describe a methodology to compute and classify the DSEMs listed in Table 2. Without loss of generality, we illustrate the methodology for type . The same procedure is used for the remaining six types.

Let M be a DSEM of type with vertex set V on a surface of Euler characteristic 0 (i.e., on torus or Klein bottle). Let and denote the set of vertices with face-sequence type (3⁶) and (3³, 4²), respectively. Here and denote the cardinality of the sets and , respectively. Then, it is easy to see that the number of triangular faces is or . Thus, if the map exists, then . Therefore, we have such that . Now we use the following steps to enumerate DSEM M for this .

Steps to enumerate DSEMs of type : Step 1:(1)Without loss of generality, let us start with a vertex having face-sequence type (3⁶). Let lk(a₁) = C₆(a₂, 1, 2, a₃, 3, 4).(2)This implies lk(a₂) = C₆(1, a₁, 4, n₂, x₁, n₁) with several choices for the triplet (n₁, x₁, n₂) in or with several choices for (see Figure 2).(3)Again among lk(a₂) and lk(4), without loss of generality, we proceed with . For each choice of (n₁, x₁, n₂) we have distinct possibility for lk(a₂). Out of these possibilities of lk(a₂), we qualify those ones which preserves the face-sequence types of vertices. The similar procedure may be adopted for lk(4) (if required). Step 2: we continuously repeat Step 1 until we do not get the links of remaining vertices from V. Step 3: the computation involved in Step 1 and Step 2 is case by case and exhaustive covering all possible scenarios. Step 4: we explore isomorphism between the maps obtained in Step 1 and Step 2, which leads to the enumeration of DSEMs of type .

(a)

(b)

Figure 2 

Illustration of the methodology. (a) DSEM type: [3⁶: 3³, 4²]₁. (b) lk(a₁) = C₆(a₂, 1, 2, a₃, 3, 4)  lk(a₂) or lk(4).

To show that two maps M₁ and M₂ are non-isomorphic, we compute the characteristic polynomials p(EG(M₁)) and p(EG(M₂)) of adjacency matrices associated the edge graphs EG(M₁) and EG(M₂) of the maps M₁ and M₂, respectively. The edge graph of M is a graph EG(M) consisting of vertices and edges of the map. Clearly, if p(EG(M₁)) ≠ p(EG(M₂)), M₁ ≇ M₂. However, if p(EG(M₁)) = p(EG(M₂)), we cannot say anything.

4. Computation and Classification of DSEMs

In this section, we compute and classify the seven types DSEMs (listed in Table 2) using the methodology given in Section 3. For the sake of computation, we consider the number of vertices ≤15.

4.1. Computation and Classification for Type

Consider the following DSEMs of type , in Figure 3, on torus and Klein bottle denoted by , for i ∈{1, …, 8}, and , for i ∈{1, …, 6}, respectively.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

Figure 3 

Doubly semiequivelar maps on torus and Klein bottle of type . (a) T_{1(3, 6)} [3⁶: 3³, 4²]₁. (b) T_{2(3, 6)} [3⁶: 3³, 4²]₁. (c) K_{1(3, 6)} [3⁶: 3³, 4²]₁. (d) K_2(3,6) [3⁶: 3³, 4²]₁. (e) T_3(4,8) [3⁶: 3³, 4²]₁. (f) T_4(4,8) [3⁶: 3³, 4²]₁. (g) K_3(4,8) [3⁶: 3³, 4²]₁. (h) K_4(4,8) [3⁶: 3³, 4²]₁. (i) K_5(4,8) [3⁶: 3³, 4²]₁. (j) T_5(4,8) [3⁶: 3³, 4²]₁. (k) T_6(5,10) [3⁶: 3³, 4²]₁. (l) T_7(5,10) [3⁶: 3³, 4²]₁. (m) K_6(5,10) [3⁶: 3³, 4²]₁. (n) K_7(5,10) [3⁶: 3³, 4²]₁. (o) T_8(5,10) [3⁶: 3³, 4²]₁.

Claim 1. For the maps above, we have the following:(a).(b).(c).(d).(e).

Proof. Let p(EG(M)) denote the characteristic polynomial of adjacency matrix associated with the edge graph of M. Then, the proof follows from the following polynomials:             

Claim 2. .

Proof. Note that , but the maps are non-isomorphic. To see this, we use geometric argument as follows: define a basis {a, b} at any vertex (for 1 ≤ i ≤ 6), where a and b are minimal nontrivial loops (i.e., nontrivial cycle with minimum number of vertices); now, if we consider , then at each , we get a and b with length 3 (for example, at , a = C₃() and b = C₃()) while in , at each , we get a of length 3 and b of length 4 (for example, at , we see that a = C₃() or a = C₃() and b = C₄()). Hence, .

4.1.1. Computation

Let M be a DSEM of type with the vertex set V. Let and denote the sets of vertices with face-sequence types (3⁶) and (3³, 4²), respectively. Then, we see that the number of triangular faces in M is or . This implies . Thus, for , we let , where . Without loss of generality, we may assume lk(a₁) = C₆(a₂, 1, 2, a₃, 3, 4). This implies lk(a₂) = C₆(a₁, 1, n₁, x₁, n₂, 4), lk(1) = C₇(a₁, a₂, [n₁, n₃, n₄, n₅, 2]), , , , , , and for some and .

Now considering lk(a₂), we see that n₁≠2 or 3 (for n₁ = 2, the set {2, 1, a₂} forms triangular face in lk(a₁) but not in lk(a₂); for n₁ = 3, we get deg(3) > 5). Similarly, we see that n₂∉{2, 3}. From these observations, we have (n₁, x₁, n₂) ∈{(5, a₃, 6), (5, a₄, 6)}. Case 1. if (n₁, x₁, n₂) = (5, a₃, 6), then lk(a₃) = C₆(a₁, 3, 6, a₂, 5, 2) or . When , considering lk(2), we have . If , then lk(1) is a cycle of length 5, a contradiction. On the other hand, if , then ; completing successively, we get , , , and . Then, we get by the map i↦, a_j↦u_j, 1 ≤ i ≤ 6, 1 ≤ j ≤ 3. When lk(a₃) = C₆(a₁, 3, 6, a₂, 5, 2), considering lk(2), we get (n₄, n₅, n₇) ∈{(4, 3, 6), (6, 3, 4), (3, 4, 6), (6, 4, 3), (3, 6, 4), (4, 6, 3)}. Observe that (3, 4, 6) ≅ (6, 3, 4) by the map (1, 5, 2) (3, 4, 6) (a₁, a₂, a₃), (3, 6, 4) ≅ (6, 4, 3) by the map (1, 5) (4, 6) (a₁, a₃), and (4, 6, 3) ≅ (6, 3, 4) by the map (1, 2, 5) (3, 6, 4) (a₁, a₃, a₂). So, we need to search only for (n₄, n₅, n₇) ∈ {(4, 3, 6), (6, 3, 4), (6, 4, 3)}. In case (n₄, n₅, n₇) = (4, 3, 6), completing successively, we get , , , , , . This gives by the map i↦, a_j↦u_j, 1 ≤ i ≤ 6, 1 ≤ j ≤ 3. Proceeding similarly as above, for (n₄, n₅, n₇) = (6, 3, 4), we get by the map i↦, a_j↦u_j, where 1 ≤ i ≤ 6, 1 ≤ j ≤ 3. For (n₄, n₅, n₇) = (6, 4, 3), , by the map i↦, a_j↦u_j, for 1 ≤ i ≤ 6, 1 ≤ j ≤ 3. Case 2. For (n₁, x₁, n₂) = (5, a₄, 6), considering lk(a₃), we get x₂ ∈{a₄, a₅}. Subcase 2.1. If x₂ = a₄, then lk(a₃) = C₆(a₁, 2, 7, a₄, 8, 3). This implies lk(a₄) = C₆(a₂, 5, 7, a₃, 8, 6) or lk(a₄) = C₆(a₂, 5, 8, a₃, 7, 6). When lk(a₄) = C₆(a₂, 5, 7, a₃, 8, 6), then, up to isomorphism, we see that (n₁₃, n₁₄, n₁₅) ∈ {(2, 1, 5), (5, 1, 2), (1, 2, 7), (7, 2, 1), (7, 5, 1)}. Now doing computation for these cases, we see the following: If (n₁₃, n₁₄, n₁₅) = (2, 1, 5), by i↦, a_j↦u_j, 1 ≤ i ≤ 8, 1 ≤ j ≤ 4. If by i↦, a_j↦u_j, 1 ≤ i ≤ 8, 1 ≤ j ≤ 4. If by i↦, a_j↦u_j, 1 ≤ i ≤ 8, 1 ≤ j ≤ 4. If by i↦, a_j↦u_j, 1 ≤ i ≤ 8, 1 ≤ j ≤ 4. If by i↦, a_j↦u_j, 1 ≤ i ≤ 8, 1 ≤ j ≤ 4. On the other hand when lk(a₄) = C₆(a₂, 5, 8, a₃, 7, 6), we get (n₁₃, n₁₄, n₁₅) ∈{(2, 1, 5), (1, 5, 8), (3, 8, 5), (5, 1, 2), (5, 8, 3), (8, 5, 1)}. If (n₁₃, n₁₄, n₁₅) = (2, 1, 5) and (1, 5, 8), then lk(7) is a cycle of length 5 and 6 respectively, which is not possible. If (n₁₃, n₁₄, n₁₅) = (3, 8, 5) and (5, 1, 2), then we see easily that lk(7) and lk(8) cannot be completed, respectively. If (n₁₃, n₁₄, n₁₅) = (5, 8, 3), then completing lk(7) we get lk(1) of length 5, again a contradiction. If by i↦, a_j↦u_j, 1 ≤ i ≤ 8, 1 ≤ j ≤ 4. Subcase 2.2. When x₂ = a₅, successively, we get lk(a₃) = C₆(a₁, 2, 7, a₅, 8, 3) and lk(a₄) = C₆(a₂, 6, 9, a₅, 10, 5). This implies lk(a₅) = C₆(a₄, 9, 7, a₃, 8, 10) or lk(a₅) = C₆(a₄, 9, 8, a₃, 7, 10). In case lk(a₅) = C₆(a₄, 9, 7, a₃, 8, 10), considering lk(1), we get (n₃, n₄, n₅) ∈ {(3, 4, 6), (3, 8, 10), (4, 3, 8), (4, 6, 9), (6, 4, 3), (6, 9, 7), (7, 9, 6), (8, 3, 4), (9, 6, 4), (10, 8, 3)}. But a small calculation shows that no map exists for these cases, except for (n₃, n₄, n₅) = (6, 4, 3). For (n₃, n₄, n₅) = (6, 4, 3), we get by the map i↦, a_j↦u_j, 1 ≤ i ≤ 10, 1 ≤ j ≤ 5. On the other hand, when lk(a₅) = C₆(a₄, 9, 8, a₃, 7, 10), considering lk(1), up to isomorphism, we get (n₃, n₄, n₅) ∈ {(3, 4, 6), (3, 8, 9), (4, 3, 8), (6, 4, 3)}. Now doing computation for these cases, we see the following: If (n₃, n₄, n₅) = (3, 4, 6), by i↦, a_j↦u_j, 1 ≤ i ≤ 10 and 1 ≤ j ≤ 5. If (n₃, n₄, n₅) = (3, 8, 9), by i↦, a_j↦u_j, 1 ≤ i ≤ 10 and 1 ≤ j ≤ 5. If (n₃, n₄, n₅) = (4, 3, 8), by i↦, a_j↦u_j, 1 ≤ i ≤ 10 and 1 ≤ j ≤ 5. If (n₃, n₄, n₅) = (6, 4, 3), by i↦, a_j↦u_j, 1 ≤ i ≤ 10 and 1 ≤ j ≤ 5. This completes computation for the number of vertices ≤15 and we obtain the following results.

4.1.2. Results

Lemma 1. Let M be a DSEM of type with number of vertices ≤15. Then, M is isomorphic to one of the following: , , , , , , , , , , , , , , and , as shown in Figure 3.

Combining Lemma 1 together with Claims 1 and 2, we get the following theorem.

Theorem 1. There are exactly 15 DSEMs of type on the surfaces of Euler characteristic 0 with number of vertices ≤15. Out of these, 8 are on the torus and remaining 7 are on the Klein bottle.

4.2. Computation and Classification for Type

Consider the following DSEMs of type , shown in Figure 4, on torus and Klein bottle denoted by , for i = 1, 2, and , respectively.

(a)

(b)

(c)

Figure 4 

Doubly semiequivelar maps on torus and Klein bottle of type . (a) K_1(6,6) [3⁶: 3³, 4²]₂. (b) T_1(6,6) [3⁶: 3³, 4²]₂. (c) T_2(6,6) [3⁶: 3³, 4²]₂.

Claim 3. .

Proof. The proof follows from the following polynomials:  

4.2.1. Computation

Let M be a map of the type with the vertex set V. Let and denote the sets of vertices with face-sequence types (3⁶) and (3³, 4²), respectively. Observe that M has the number of edges = , number of triangular faces = , and number of quadrangular faces = . Now by the Euler characteristic equation, we get . This implies . Also, considering the number of quadrangular faces, it is evident that the cardinality of both the sets should be positive even integer. Thus, for |V| ≤ 15, we let , where for k ≤ 3.

Without loss of generality, assume lk(a₁) = C₆(a₂, a₃, a₄, a₅, 1, 2). This implies lk(1) = C₇(a₁, a₅, [n₁, n₂, n₃, n₄, 2]), lk(2) = C₇(a₂, a₁, [1, n₃, n₄, n₆, n₅]) and lk(a₂) = C₆(a₁, a₃, x₁, x₂, 3, 2) for some n₁, …, n₆ ∈ V_m and x₁, x₂ ∈ V_l. It is easy to see that (x₁, x₂) ∈{(a₅, a₄), (a₆, a₅)}. Case 1. When (x₁, x₂) = (a₅, a₄), then successively we get lk(a₂) = C₆(a₁, a₃, a₅, a₄, 3, 2), lk(a₅) = C₆(a₁, a₄, a₂, a₃, 4, 1), lk(a₃) = C₆(a₄, a₁, a₂, a₅, 4, 5) and lk(a₄) = C₆(a₃, a₁, a₅, a₂, 3, 5). Now considering lk(1), we see that (n₂, n₃, n₄) has no value for the V so that the links of remaining vertices can be completed. So, (x₁, x₂) ≠ (a₅, a₄). Case 2. When (x₁, x₂) = (a₆, a₅), then successively we get lk(a₂) = C₆(a₁, a₃, a₆, a₅, 3, 2), lk(a₅) = C₆(a₁, a₄, a₆, a₂, 3, 1), lk(a₄) = C₆(a₃, a₁, a₅, a₆, 5, 4), lk(a₃) = C₆(a₄, a₁, a₂, a₆, 6, 4), lk(a₆) = C₆(a₃, a₂, a₅, a₄, 5, 6). Considering lk(1), it is easy to see that (n₂, n₃, n₄) ∈{(4, 5, 6), (4, 6, 5), (5, 4, 6), (5, 6, 4), (6, 4, 5), (6, 5, 4)}. Observe that (4, 5, 6) ≅ (6, 4, 5) by the map (1, 3, 2) (5, 6, 4) (a₁, a₅, a₂) (a₃, a₄, a₆), (4, 5, 6) ≅ (5, 6, 4) by the map (1, 2, 3) (4, 6, 5) (a₁, a₂, a₅) (a₃, a₆, a₄), and (5, 4, 6) ≅ (6, 5, 4) by the map (1, 3) (4, 6) (a₁, a₂) (a₄, a₆). So, we search for (n₂, n₃, n₄) ∈ {(4, 5, 6), (4, 6, 5), (5, 4, 6)}. Now doing computation for these cases, we see the following: If by the map i↦, a_i↦u_i, 1 ≤ i ≤ 6. If by the map i↦, a_i↦u_i, 1 ≤ i ≤ 6. If by the map i↦, a_i↦u_i, 1 ≤ i ≤ 6. This completes computation for ≤15 vertices. From this we get following results.

4.2.2. Results

Lemma 2. Let M be a DSEM of type on the surfaces of Euler characteristic 0 with ≤15 vertices. Then, M is isomorphic to one of , , and , given in Figure 4.

Combining Lemma 2 with Claim 3, we get the following theorem.

Theorem 2. There are exactly 3 non-isomorphic DSEMs of type with number of vertices ≤15. Out of these, 2 are on torus and the remaining one is on Klein bottle.

4.3. Computation and Classification for Type [3⁶: 3², 4, 3, 4]

Consider the following DSEM of type [3⁶: 3², 4, 3, 4], given in Figure 5 on Klein bottle, denoted by K_1(2,12)[3⁶: 3², 4, 3, 4].

Figure 5 

Doubly semiequivelar maps on Klein bottle of type K_1(2,12) [3⁶: 3², 4, 3, 4].

4.3.1. Computation

Let M be a map of the type [3⁶: 3², 4, 3, 4] with the vertex set V. Let and denote the sets of vertices with face-sequence types (3⁶) and (3², 4, 3, 4), respectively. It is easy to see that . Thus, for |V| ≤ 15, we let , where . Without loss of generality, we assume lk(a₁) = C₆(1, 2, 3, 4, 5, 6). Then, successively, we have lk(1) = C₇(a₁, [2, n₁, n₂], [n₃, n₄, 6]), lk(2) = C₇(a₁, [1, n₂, n₁], [n₅, n₆, 3]), lk(3) = C₇(a₁, [2, n₅, n₆], [n₇, n₈, 4]), lk(4) = C₇(a₁, [3, n₇, n₈], [n₉, n₁₀, 5]), lk(5) = C₇(a₁, [4, n₉, n₁₀], [n₁₁, n₁₂, 6]), lk(6) = C₇(a₁, [1, n₃, n₄], [n₁₂, n₁₁, 5]), where n_i ∈ V_m for 1 ≤ i ≤ 12. Now considering lk(1), we see that n₁ ∈{4, 5, 7}. Case 1. If n₁ = 4, then successively, we see that n₂ = 5, n₃ = 7, and n₄ = 8; now considering lk(5) and lk(1), we get two quadrangular faces which share more than one vertex, which is not allowed. So, n₁ ≠ 4. Case 2. If n₁ = 5, then successively, we get n₂ = 4, n₃ = 7, n₄ = 8, n₁₂ = 9, and n₁₁ = 10. Now completing lk(1), lk(6), lk(5), lk(2), lk(3), lk(4), lk(7), and lk(10), we see that lk(a₂) = C₆(7, 8, 11, 10, 9, 12) or lk(a₂) = C₆(7, 8, 9, 10, 11, 12). If lk(a₂) = C₆(7, 8, 9, 10, 11, 12), then lk(8) is a cycle of length 5, a contradiction. If lk(a₂) = C₆(7, 8, 11, 10, 9, 12), then completing successively, we get M ≅ K_1(2,12)[3⁶: 3², 4, 3, 4] by the map i↦, a_j↦u_j, 1 ≤ i ≤ 12, 1 ≤ j ≤ 2. Case 3. If n₁ = 7, then we get (n₂, n₃) ∈ {(8, 3), (8, 4), (8, 9)}. For (n₂, n₃) = (8, 3), n₄ = 4 and lk(1) = C₇(a₁, [2, 7, 8], [3, 4, 6]). Now considering successively lk(3) and lk(1), we see two quadrangular faces which share more than one vertex, which is not allowed. Hence, (n₂, n₃) ≠ (8, 3). For (n₂, n₃) = (8, 4), completing successively lk(1), lk(4), lk(5), lk(6), lk(3), lk(2), lk(11), and lk(8), we see that lk(a₂) = C₆(7, 8, 9, 10, 11, 12) or lk(a₂) = C₆(7, 8, 9, 12, 11, 10). If lk(a₂) = C₆(7, 8, 9, 10, 11, 12), then lk(7) is a cycle of length 5, a contradiction. If lk(a₂) = C₆(7, 8, 9, 12, 11, 10), completing successively, we get M ≅ K_1(2,12)-[3⁶: 3², 4, 3, 4] via 1↦, 2↦, 3↦, 4↦, 5↦, 6↦, 7↦, 8↦, 9↦, 10↦, 11↦, 12↦, a₁↦u₂, a₂↦u₁. If (n₂, n₃) = (8, 9), n₄ = 10. This implies lk(1) = C₇(a₁, [2, 7, 8], [9, 10, 6]) and lk(6) = C₇(a₁, [1, 9, 10], [n₁₂, n₁₁, 5]) for (n₁₁, n₁₂) ∈ {(3, 2), (11, 4), (11, 7), (11, 12)}. A small calculation shows no map exists for (n₁₁, n₁₂) ∈ {(11, 4), (11, 7), (11, 12)}. For (n₁₁, n₁₂) = (3, 2), completing successively, we get M ≅ K_1(2,12)[3⁶: 3², 4, 3, 4] via 1↦, 2↦, 3↦, 4↦, 5↦, 6↦, 7↦, 8↦, 9↦, 10↦, 11↦, 12↦, a₁↦u₁, a₂↦u₂. This completes computation of the DSEM for ≤15. This gives the following result.

4.3.2. Result

Theorem 3. There exists a unique DSEM of type [3⁶: 3², 4, 3, 4] on the surfaces of Euler characteristic 0 for ≤15 vertices. This is K_1(2,12)[3⁶: 3², 4, 3, 4] on Klein bottle, given in Figure 5.

4.4. Computation and Classification for Type

Consider the following DSEM of type , shown in Figure 6, on torus denoted by .

Figure 6 

Doubly semiequivelar map on torus of type T_1(4,8) [3³, 4²: 3², 4, 3, 4]₁.

4.4.1. Computation

Let M be a map of the type with the vertex set V. Let and denote the sets of vertices with face-sequence types (3³, 4²) and (3², 4, 3, 4), respectively. It is easy to see that and is multiple of 4. Therefore, for |V| ≤ 15, we let , where . Assume that lk(a₁) = C₇(2, 3, [4, 5, a₂, 6, 1]). Then, lk(a₂) = C₇(7, 8, [5, 4, a₁, 1, 6]). This implies lk(2) = C₇(a₁, [1, x₁, x₂], [n₁, n₂, 3]) or lk(2) = C₇(a₁, [3, x₁, x₂], [n₁, n₂, 1]), for and . In the first case of lk(2), considering lk(1), we see three quadrangular faces incident at 1, which is not allowed. On the other hand, when lk(2) = C₇(a₁, [3, x₁, x₂], [n₁, n₂, 1]), we get x₁ = a₃, x₂ = a₄, and (n₁, n₂) ∈ {(5, 8), (7, 8), (8, 5), (8, 7)}.

For (n₁, n₂) = (7, 8), considering lk(1) and lk(2), we see that lk(a₃) = C₇(1, 8, [3, 2, a₄, n₃, 6]) or lk(a₃) = C₇(1, 6, [3, 2, a₄, n₃, 8]), but for both the cases of lk(a₃), we get no suitable value for n₃ in . So, (n₁, n₂) ≠ (7, 8).

For (n₁, n₂) = (8, 5), completing lk(2) and lk(1) and proceeding, as in previous case, we see that lk(3) cannot be completed.

For (n₁, n₂) = (8, 7), considering lk(2) and lk(1), we see that lk(7) cannot be completed.

For (n₁, n₂) = (5, 8), successively, we get lk(2) = C₇(a₁, [3, a₃, a₄], [5, 8, 1]), lk(5) = C₇(a₄, [4, a₁, a₂], [8, 1, 2]), and lk(1) = C₇(a₃, [6, a₂, a₁], [2, 5, 8]). Then, lk(a₃) = C₇(1, 8, [3, 2, a₄, 7, 6]) or lk(a₃) = C₇(1, 6, [3, 2, a₄, 7, 8]).

When lk(a₃) = C₇(1, 8, [3, 2, a₄, 7, 6]), completing lk(a₄), we see that lk(7) cannot be completed.

When lk(a₃) = C₇(1, 6, [3, 2, a₄, 7, 8]), completing successively, we get by the map i↦, a_j↦u_j, 1 ≤ i ≤ 8, 1 ≤ j ≤ 4. Thus, the computation is completed for ≤15 vertices. This leads to the following result.

4.4.2. Result

Theorem 4. There exists a unique DSEM of type with number of vertices ≤15. This is on torus, shown in Figure 6.

4.5. Computation and Classification for Type

Consider the following DSEM of type , shown in Figure 7, on Klein bottle denoted by .

Figure 7 

Doubly semiequivelar map of type on Klein bottle.

4.5.1. Computation

Let M be a map of the type with the vertex set V. Let and denote the sets of vertices with face-sequence types (3³, 4²) and (3², 4, 3, 4), respectively. Then, we see easily that for . Thus, for , we let , where for k ≤ 3. Assume that, without loss of generality, lk(a₁) = C₇(a₃, 3, [4, 5, a₂, 1, 2]). This implies lk(a₂) = C₇(x₁, n₁, [5, 4, a₁, 2, 1]) or lk(a₂) = C₇(x₁, n₁, [1, 2, a₁, 4, 5]) for and .

If lk(a₂) = C₇(x₁, n₁, [5, 4, a₁, 2, 1]), then a small calculation shows that no such map exists for the given V. On the other hand, if lk(a₂) = C₇(x₁, n₁, [1, 2, a₁, 4, 5]), then we have x₁ = a₄, n₁ = 6. This implies lk(2) = C₇(a₃, [a₁, a₂, 1], [x₂, x₃, n₂]), where (x₂, x₃, n₂) ∈ {(a₅, a₄, 5), (a₅, a₄, 6)}. If (x₂, x₃, n₂) = (a₅, a₄, 5), then lk(a₃) = C₇(a₁, 2, [5, n₃, a₆, n₄, 3]). Now considering lk(5), we see three quadrangular faces incident at 5, which is not allowed. If (x₂, x₃, n₂) = (a₅, a₄, 6), lk(2) = C₇(a₃, [a₁, a₂, 1], [a₅, a₄, 6]). This implies lk(a₃) = C₇(a₁, 2, [6, n₃, a₆, n₄, 3]), where (n₃, n₄) ∈ {(1, 5), (4, 1), (4, 5)}. In case (n₃, n₄) = (4, 1) and (4, 5), we see, respectively, lk(1) and lk(4) cannot be completed. If by the map i↦, a_i↦u_i, 1 ≤ i ≤ 6. Thus, the computation is completed. Then, we obtain the following result.

4.5.2. Result

Theorem 5. There exists a unique DSEM of type with number of vertices ≤15. This is on Klein bottle, shown in Figure 7.

4.6. Computation and Classification for Type

Consider the DSEMs of type , shown in Figure 8, on torus and Klein bottle denoted by , for i ∈ {1, …, 6}, and , for i ∈ {1, …, 3}, respectively.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

Figure 8 

Doubly semiequivelar maps of type on torus and Klein bottle. (a) K_1(3,6) [3³, 4²: 4⁴]₁. (b) T_1(3,6) [3³, 4²: 4⁴]₁. (c) T_2(3,6) [3³, 4²: 4⁴]₁. (d) K_2(4,8) [3³, 4²: 4⁴]₁. (e) T_3(4,8) [3³, 4²: 4⁴]₁. (f) T_4(4,8) [3³, 4²: 4⁴]₁. (g) T_5(5,10) [3³, 4²: 4⁴]₁. (h) T_6(5,10) [3³, 4²: 4⁴]₁. (i) K_4(5,10) [3³, 4²: 4⁴]₁. (j) T_7(5,10) [3³, 4²: 4⁴]₁. (k) K_3(5,10) [3³, 4²: 4⁴]₁.

Claim 4. For the maps above, we have the following:(a).(b).(c).(d)-.

Proof. The proof follows by considering the following polynomials:         

4.6.1. Computation

Let M be a map of the type with the vertex set V. Let and denote the sets of vertices with face-sequence types (4⁴) and (3³, 4²), respectively. Then, counting the number of quadrangular faces in terms of and we see easily that . Thus, for |V| ≤ 15, we let such that . Assume that, without loss of generality, . This implies lk(a₂) = C₈(a₁, 2, 3, n₁, x₁, n₂, 4, 5) for and . Observe that x₁ ∈ {a₃, a₄}. Case 1. When x₁ = a₃, then (n₁, n₂) ∈ {(1, 6), (6, 1)}. If (n₁, n₂) = (6, 1), then lk(a₂) = C₈(a₁, 2, 3, 6, a₃, 1, 4, 5) and lk(a₃) = C₈(a₂, 3, 6, 5, a₁, 2, 1, 4). This implies lk(1) = C₇(n₃, n₄, [2, a₁, a₃, a₂, 4]). It is easy to see that (n₃, n₄) ∈ {(3, 6), (5, 6), (6, 3), (6, 5)}. But for these values of (n₃, n₄), we see easily that no map exists. On the other hand, if (n₁, n₂) = (1, 6), then lk(a₂) = C₈(a₁, 2, 3, 1, a₃, 6, 4, 5), lk(a₃) = C₈(a₂, 3, 1, 2, a₁, 5, 6, 4). This implies lk(1) = C₇(n₃, n₄, [2, a₁, a₃, a₂, 3]), for (n₃, n₄) ∈ {(4, 5), (4, 6), (5, 4), (5, 6), (6, 4), (6, 5)}. Observe that (5, 6) ≅ (4, 5) by the map (1, 3, 2) (4, 5, 6) (a₁, a₃, a₂), (6, 4) ≅ (4, 5) by the map (1, 2, 3) (4, 6, 5) (a₁, a₂, a₃), and (6, 5) ≅ (4, 6) by the map (2, 3) (4, 5) (a₁, a₂). Thus, we search for (n₃, n₄) ∈ {(4, 5), (4, 6), (5, 4), (5, 6)}. Now doing computation for these cases, we see the following: If by the map i↦, a_j↦u_j, 1 ≤ i ≤ 6, 1 ≤ j ≤ 3. If by the map i↦, a_j↦u_j, 1 ≤ i ≤ 6, 1 ≤ j ≤ 3. If by the map i↦, a_j↦u_j, 1 ≤ i ≤ 6, 1 ≤ j ≤ 3. Case 2. When x₁ = a₄, then considering lk(a₃) = C₈(a₁, 2, 1, n₃, x₂, n₄, 6, 5), we get x₂ ∈ {a₄, a₅}. Subcase 2.1. If x₂ = a₄, then (n₃, n₄) ∈ {(7, 8), (8, 7)}. If (n₃, n₄) = (8, 7), (n₅, n₆) ∈ {(4, 5), (5, 4), (5, 6), (6, 5), (6, 7), (7, 3), (7, 6)}. If (n₅, n₆) = (4, 5), then considering successively lk(1), lk(5), and lk(8), we see that deg(4) > 5, a contradiction. If (n₅, n₆) = (5, 4), then considering successively lk(1), lk(5), lk(2), and lk(6), we get lk(7) of length 5, a contradiction. Proceeding similarly for the rest of the cases of (n₅, n₆), we see easily that no map exists. On the other hand, if (n₃, n₄) = (7, 8), then lk(1) = C₇(n₅, n₆, [2, a₁, a₃, a₄, 7]), where (n₅, n₆) ∈ {(4, 5), (4, 8), (5, 4), (5, 6), (6, 5), (6, 8), (8, 4), (8, 6)}. But (5, 4) ≅ (4, 8) by the map (1, 3) (4, 6) (a₂, a₃), (5, 6) ≅ (4, 5) by the map (1, 7) (2, 3) (4, 5) (6, 8) (a₁, a₂) (a₃, a₄), (6, 8) ≅ (4, 5) by the map (1, 2, 3, 7) (4, 8, 6, 5) (a₁, a₂, a₄, a₃), (8, 4) ≅ (4, 5) by the map (1, 3) (2, 7) (4, 6) (5, 8) (a₁, a₄) (a₂, a₃), and (8, 6) ≅ (6, 5) by the map (1, 7) (2, 3) (4, 5) (6, 8) (a₁, a₂) (a₃, a₄). So, we search for (n₅, n₆) ∈ {(4, 5), (4, 8), (6, 5)}. Now doing computation for these cases, we see the following: If by the map i↦, a_j↦u_j, 1 ≤ i ≤ 8, 1 ≤ j ≤ 4. If by the map i↦, a_j↦u_j, 1 ≤ i ≤ 8, 1 ≤ j ≤ 4. If by the map i↦, a_j↦u_j, 1 ≤ i ≤ 8, 1 ≤ j ≤ 4. Subcase 2.2. When x₂ = a₅, then considering lk(a₄), we see that (n₅, n₆) ∈ {(9, 10), (10, 9)}. If (n₅, n₆) = (10, 9), (n₇, n₈) ∈ {(3, 7), (4, 5), (4, 8), (5, 4), (5, 6), (6, 5), (6, 10), (7, 10), (10, 6), (10, 7)}. But a small calculation shows that no map exists for these values of (n₇, n₈). If $(n5, n6) = (9,10)$, then we get lk(1) = C₇(n₇, n₈, [2, a₁, a₃, a₅, 9]). Then, up to isomorphism, we get (n₇, n₈) ∈ {(3, 7), (4, 5), (5, 4), (6, 5)}. Doing computation for these cases, we see the following: If by the map i↦, a_j↦u_j, 1 ≤ i ≤ 10, 1 ≤ j ≤ 5. If by the map i↦, a_j↦u_j, 1 ≤ i ≤ 10, 1 ≤ j ≤ 5. If by the map i↦, a_j↦u_j, 1 ≤ i ≤ 10, 1 ≤ j ≤ 5. If by the map i↦, a_j↦u_j, 1 ≤ i ≤ 10, 1 ≤ j ≤ 5. If by the map i↦, a_j↦u_j, 1 ≤ i ≤ 10, 1 ≤ j ≤ 5. This completes the computation and we get the following results.

4.6.2. Results

Lemma 3. Let M be a DSEM of type with number of vertices ≤15. Then, M is isomorphic to one of the following: , , , , , , , , , , and , as shown in Figure 8.

Combining Lemma 3 together with Claim 4, we get the following theorem.

Theorem 6. There are exactly 11 non-isomorphic DSEMs of type on the surfaces of Euler characteristic 0 with ≤15 vertices. Out of these, 7 are on torus and remaining 4 are on Klein bottle.

4.7. Computation and Classification for Type

Consider the following DSEMs of type , shown in Figure 9, on torus and Klein bottle denoted by , for i = 1, 2, and , respectively.

(a)

(b)

(c)

Figure 9 

Doubly semiequivelar maps of type on torus and Klein bottle. (a) K_1(6,6) [3³, 4²: 4⁴]₂. (b) T_1(6,6) [3³, 4²: 4⁴]₂. (c) T_2(6,6) [3³, 4²: 4⁴]₂.

Claim 5. .

Proof. See the following polynomials:  

4.7.1. Computation

Let M be a map of the type with the vertex set V. Let and denote the sets of vertices with face-sequence types (4⁴) and (3³, 4²), respectively. Then, it is easy to see that M has the number of quadrangular faces or . This implies for . Therefore, for |V| ≤ 15, we let , where for k ≤ 3. Without loss of generality, we may assume lk(1) = C₇(3, 4, [5, a₁, a₂, a₃, 2]). Then, lk(a₂) = C₈(a₃, 2, 1, 5, a₁, a₆, a₅, a₄), lk(a₁) = C₈(a₂, 1, 5, n₁, x₁, x₂, a₆, a₅) for n₁ ∈ V_m and x₁, x₂ ∈ V_l. Observe that (n₁, x₁, x₂) ∈{(2, a₃, a₄), (3, a₄, a₃), (6, a₄, a₃)}. Case 1. When (n₁, x₁, x₂) = (3, a₄, a₃), then considering successively lk(a₁), lk(a₄), lk(a₅), lk(3), and lk(4), we see that lk(5) cannot be completed. So, (n₁, x₁, x₂) ≠ (2, a₄, a₃). Case 2. When (n₁, x₁, x₂) = (6, a₄, a₃), then lk(a₁) = C₈(a₂, 1, 5, 6, a₄, a₃, a₆, a₅) and lk(a₄) = C₈(a₁, 5, 6, n₂, a₅, a₂, a₃, a₆) for n₂ ∈{3, 4, 7}. If n₂ = 3, then considering successively lk(a₄), lk(a₅), lk(a₆), lk(a₃), lk(3), and lk(2), we see lk(4) cannot be completed. If n₂ = 4, then considering successively lk(a₄), lk(a₅), lk(a₆), lk(a₃), and lk(4), as in previous case, we see that lk(5) cannot be completed. If n₂ = 7, then considering successively lk(a₄), lk(a₅), and lk(a₆), we see that lk(a₃) cannot be completed. So, (n₁, x₁, x₂) ≠ (6, a₄, a₃). Case 3. If (n₁, x₁, x₂) = (2, a₃, a₄), then successively completing lk(a₁), lk(a₃), lk(2), and lk(5), we get lk(4) = C₇(1, 5, [6, x₅, x₄, x₃, 3]) for (x₃, x₄, x₅) ∈ {(a₅, a₄, a₆), (a₆, a₄, a₅), (a₄, a₅, a₆), (a₆, a₅, a₄), (a₄, a₆, a₅), (a₅, a₆, a₄)}. Note that (a₆, a₅, a₄) ≅ (a₅, a₄, a₆) by the map (1, 5, 2) (3, 4, 6) (a₁, a₃, a₂) (a₄, a₅, a₆), (a₄, a₆, a₅) ≅ (a₅, a₄, a₆) by the map (1, 2, 5) (3, 6, 4) (a₁, a₂, a₃) (a₄, a₆, a₅), and (a₅, a₆, a₄) ≅ (a₄, a₅, a₆) by the map (2, 5) (3, 4) (a₁, a₃) (a₄, a₆). Thus, we do computation for (x₃, x₄, x₅) ∈ {(a₅, a₄, a₆), (a₆, a₄, a₅), (a₄, a₅, a₆)}. This gives the following: If by the map i↦, a_i↦u_i, 1 ≤ i ≤ 6. If by the map i↦, a_i↦u_i, 1 ≤ i ≤ 6. If by the map i↦, a_i↦u_i, 1 ≤ i ≤ 6. Thus, the computation is completed and we get the following results.

4.7.2. Results

Lemma 4. Let M be a DSEM of type with number of vertices ≤15. Then, M is isomorphic to , or , as shown in Figure 9.

Combining Lemma 4 together with Claim 5, we get the following theorem.

Theorem 7. There are exactly 3 non-isomorphic DSEMs of type on the surfaces of Euler characteristic 0 with ≤15 vertices. Out of these, two are on torus and the remaining one is on Klein bottle.

5. Summary

From Theorems 1–7, we get the following.

Theorem 8. There are at least 35 non-isomorphic DSEMs on the surfaces of Euler characteristic 0 with ≤15 vertices. Out of these, 20 are on the torus and remaining 15 are on the Klein bottle.

A tabular form of the results obtained here is presented in Table 1.

6. Discussion

In [16], the authors constructed infinite series of semiequivelar maps on torus and Klein bottle from equivelar maps by using elementary map operations: truncation and subdivision (these operations do not affect the symmetry of a map). Here, we present infinite series of the seven type doubly semiequivelar maps for torus, and one can explore similarly for Klein bottle.

Infinite series of DSEMs of types , , , , and [3⁶: 3³, 4²] are constructed from infinite series of semiequivelar map of type [4⁴] by subdividing the quadrangular faces as shown in Figures 10–14, respectively. Infinite series of DSEM of type [3⁶: 3², 4, 3, 4] is obtained from an infinite series of semiequivelar map of type [6³] by subdividing the hexagonal faces (by introducing a new vertex and joining it to the six vertices of the face by an edge) as shown in Figure 15.

Figure 10 

DSEM of type-.

Figure 11 

DSEM of type-.

Figure 12 

DSEM of type-.

Figure 13 

DSEM of type-.

Figure 14 

DSEM of type-.

Figure 15 

DSEM of type-.

Although, we present infinite series of DSEM of type [3³, 4²: 3², 4, 3, 4] (see Figure 16), we do not know whether this DSEM can be obtained from any semiequivelar map by the above elementary map operations. This observation leads to the following question.

Figure 16 

DSEM of type-.

Question 1. Can we obtain every doubly semiequivelar map (corresponding to the 2-uniform tilings) on torus and Klein bottle from semiequivelar maps (corresponding to the Archimedean tilings) by applying finite sequence of map operations on the same surface?

6.1. Infinite Series of DSEMs on Torus

If we study group structures associated to the maps, we see that DSEMs, obtained here, on torus are 2-uniform. For example, in case of type , we see that the groups G₁ = , G₂ = , G₃ = , G₄ = , G₅ = , G₆ = , G₇ = , and G₈ =  act on the maps , , , , , , , and , respectively, such that under the action, the maps have two orbits of vertices. Similarly, we can easily find a group for the DSEMs of types , and on torus, under which the maps have two orbits of vertices. However, this fact does not hold for the DSEMs on Klein bottle. For example, if we let , we get no automorphism which sends to . This can be seen as follows: suppose there is such that f() = . Then, considering lk() and lk(), we see, either f() =  or f() = . In the first case, when f() = , we get f(u₂) = u₃, f(u₁) = u₁, f() = , f() = , f() = , and f() = . Now if we see lk(u₁), we get a contradiction of the facts f(u₁) = u₁ and f() = , as  ∉ lk(u₁). So, f() ≠ . Similarly, we see that f() ≠ . Combining these, we see that f() ≠ . This shows that is not 2-uniform. This observation leads to ask the following question.

Question 2. Are the doubly semiequivelar maps (corresponding to the 2-uniform tilings) on torus 2-uniform?

7. Conclusions

In this article, the notion of doubly semiequivelar maps (DSEMs) has been introduced for the first time. A methodology has been presented to enumerate doubly semiequivelar maps on torus and Klein bottle corresponding to the 2-uniform tilings , , , , , , and . The methodology has been demonstrated to enumerate the DSEMs on at most 15 vertices. The enumeration provides at least 35 non-isomorphic DSEMs on the surfaces of Euler characteristic zero, and out of these, 20 are on torus and remaining 15 are on Klein bottle. Further, infinite series of these types DSEMs have been constructed. We know that a study of maps becomes more significant when certain symmetry involves; in view of this, the notion of 2-uniform maps (parallel to the notion of vertex-transitive maps for equivelar or semiequivelar maps) has been introduced. During computation, it has been found that all the maps obtained on torus are 2-uniform, which does not hold in case of DSEMs on Klein bottle. This motivates us to explore the fact whether all the DSEMs on torus are 2-uniform. In the literature, vertex-transitive maps have been studied extensively. It would be interesting to study 2-uniform maps not only for torus and Klein bottle but also for other close surfaces and to explore the analog notions of vertex-transitive maps for 2-uniform maps.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The research work of the first author is partially supported by IIIT Allahabad.