Semi-equivelar maps on the torus Dipendu Maity Department of - - PowerPoint PPT Presentation

Semi-equivelar maps on the torus

Dipendu Maity

Department of Mathematics, Indian Institute of Science, Bangalore, India

October 3, 2016

Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 1 / 18

Objective

• What types of maps exist on the torus?
• What is the status of the classification of these maps?
• Are all these maps vertex-transitive?

Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 2 / 18

Eembedding and Map

Graph embedding on surfaces : An embedding of a graph G in a surface S is an one-one mapping i : G → S.

• A map M is an embedding of a connected finite simple graph G into a

surface S in which closure of a connected components of S \ G is homeomorphic to closed 2-disk.

• The components are called faces of M and each face is a n-gon (n ≥ 3).

So, M =: (V , E, F).

v1 v2 v3 v4 v5 v6 v7 v1 v3 v4 v5 v6 v7 v1 v2 v3 Map on the torus Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 3 / 18

Regular, Semiregular, Equivelar and Semi-equivelar map

Regular map : A map is said to be regular if the symmetry group of the map acts transitively on the flags (an incident vertex-edge-face triple) of the map. Semiregular map : A map is said to be semiregular if the symmetry group of the map acts transitively on the set of incident vertex-edge pairs

• f the map.

Equivelar map : A map M in which each face of M is a p-gon and each vertex belongs to exactly q faces is called {pq}-equivelar map. Semi-equivelar map : A map M is said to be a semi-equivelar map of type {ap, bq, . . . , mr} if face sequence of each vertex of M is {ap, bq, . . . , mr}.

• Class of regular map ⊂ Class of Equivelar map.
• Class of semiregular map ⊂ Class of Semi-Equivelar map.

Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 4 / 18

Regular and Semi-regular maps on the surfaces of χ = 0

Proposition : If {pn1

1 , . . . , pnk k } satisfies any of the following two properties

then {pn1

1 , . . . , pnk k } can not be the type of any semi-equivelar map on a

surface.

1

There exists i such that ni = 1, pi is odd, pj = pi for all j = i and pi−1 = pi+1.

2

There exists i such that ni = 2, pi is odd and pj = pi for all j = i. Corollary : Let X be a semi-equivelar map on a surface M. If M is the torus or the Klein bottle then the type of X is {36}, {44}, {63}, {33, 42}, {32, 4, 3, 4}, {3, 6, 3, 6}, {34, 6}, {4, 82}, {3, 122}, {4, 6, 12}, {3, 4, 6, 4}.

• The well known 11 types of normal tilings of the plane suggest the

possible types of semi-equivelar maps on the torus.

Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 5 / 18

Semi-equivelar maps on torus for vertices ≤ 15

• Among 11 types {36}, {44}, {63} are studied on the torus by Altshuler

[1], Brehm and K¨ uhnel [2], Kurth [6], Negami [8], Datta and Upadhyay [4].

• In 2015, the classification of remaining eight types {33, 42}, {32, 4, 3, 4},

{3, 6, 3, 6}, {34, 6}, {4, 82}, {3, 122}, {4, 6, 12}, {3, 4, 6, 4} are being attempted by Tiwari and Upadhyay [9]. Proposition : In [9], there are exactly 11 semi-equivelar maps of types {33, 42}, {32, 4, 3, 4}, {3, 6, 3, 6}, {34, 6}, {4, 82}, {3, 122}, {4, 6, 12}, {3, 4, 6, 4} with ≤ 15 vertices on the surfaces of Euler characteristic 0. Six

• f these maps are orientable and five are non-orientable.

Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 6 / 18

Semi-equivelar maps on the torus : maps of type {33, 42}

Let M be a map of type {33, 42} on the torus. We define three paths in M through each vertex. Definition 1 : Let P1 := P(. . . , ui−1, ui, ui+1, . . . ) be a path in edge graph

• f M. We say P1 of type A1 if all the triangles incident with an inner

(degree two in P1) vertex ui lie on one side and all quadrangles incident with ui lie on the other side of the portion of P1 for all i. Definition 2 : Let P2 := P(. . . , vi−1, vi, vi+1, . . . ) be a path in edge graph

• f M for which vi, vi+1 be two consecutive inner vertices of P2 or an

extended path of P2. Then we say P2 of type A2 if lk(vi) = C(a, vi−1, b, c, vi+1, d, e) implies lk(vi+1) = C(a0, vi+2, b0, d, vi, c, p) and lk(vi) = C(x, vi+1, z, l, vi−1, k, m) implies lk(vi+1) = C(l, vi, m, x, vi+2, g, z). Definition 3 : Let P3 := P(. . . , wi−1, wi, wi+1, . . . ) be a path in edge graph

• f M for which wi, wi+1 are two inner vertices of P3 or an extended path
• f P3. Then we say P3 of type A3 if lk(wi) = C(a, wi−1, b, c, d, wi+1, e)

implies lk(wi+1) = C(a1, wi+2, b1, p, e, wi, d) and lk(wi) = C(a2, wi+1, b2, p, e, wi−1, d) implies lk(wi+1) = C(p, wi, d, a2, z1, wi+2, b2).

Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 7 / 18

A2 A3 A1 v

Paths of types A1, A2, A3 through vertex v

Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 8 / 18

• Let Q be a maximal path of type At for some t ∈ {1, 2, 3}. Then, there

exists an edge e such that Q ∪ e is a cycle of type At and non-contractible.

ui−1 ui ur ur−1 ur−2 ur−3 ur−4 ur−5 ur−6 ui+1 ui+2 ui+3 ui+4 ui+5 ui+6 ui+7 wi−1 wi wi+1 wi+2 wi+3 wr−8 wr−9 wr−10

Cycle of type A2

Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 9 / 18

• The map M has a T(r, s, k) planar polyhedral representation.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v1 v2 v3 v4 vk vk+1 vk+2 vk+3 vr−2 vr−1 vr v1 w1 w2 w3 w4 wk wk+1 wk+2 wk+3 wr−2 wr−1 wr w1 x1 x2 x3 x4 xk xk+1 xk+2 xk+3 xr−2 xr−1 xr x1 z1 z2 z3 z4 zk zk+1 zk+2 zk+3 zr−2 zr−1 zr z1 vk+1 vk+2 vk+3 vk+4 vn v1 v2 v3 vk−2 vk−1 vk vk+1 Figure : T(r, 4, k)

• Let M be a map of type {33, 42} on the torus. Then, the cycles of type

A1 have unique length and the cycles of type A2 have at most two different lengths.

• The {33, 42}-maps of the form T(r, s, k) exist if and only if the

following holds : (i) s ≥ 2 even, (ii) r ≥ 3, (iii) n = rs ≥ 10, (iv) 2 ≤ k ≤ r − 3 if s = 2 & 0 ≤ k ≤ r − 1 if s ≥ 4.

Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 10 / 18

v1 v2 v3 v4 v5 v6 v7 v1 w1 w2 w3 w4 w5 w6 w7 w1 x1 x2 x3 x4 x5 x6 x7 x1 z1 z2 z3 z4 z5 z6 z7 z1 v4 v5 v6 v7 v1 v2 v3 v4 Figure 1 : T(7, 4, 3) v1 v2 v3 v4 v5 v6 v7 v1 w1 w2 w3 w4 w5 w6 w7 w1 x1 x2 x3 x4 x5 x6 x7 x1 z1 z2 z3 z4 z5 z6 z7 z1 v4 v5 v6 v7 v1 v2 v3 v4 Figure 2 : T(7, 4, 3)

• Let Ti be a map on ni vertices and n1 = n2. Then, T1 ∼

= T2 if and only if they have cycles of types A1, A2, A3, A4 of same length.

u1 u2 u3 u4 u5 u6 u7 u1 u8 u9 u10 u11 u12 u13 u14 u8 u3 u4 u5 u6 u7 u1 u2 u3 Figure 3 : T(7, 2, 2) : O1 v1 v2 v3 v4 v5 v6 v7 v1 v8 v9 v10 v11 v12 v13 v14 v8 v5 v6 v7 v1 v2 v3 v4 v5 Figure 4: T(7, 2, 4) : O2 w1 w2 w3 w4 w5 w6 w7 w1 w8 w9 w10 w11 w12 w13 w14 w8 w4 w5 w6 w7 w1 w2 w3 w4 Figure 5 : T(7, 2, 3) : O3 v5 v4 v3 v2 v1 v7 v6 v5 v12 v11 v10 v9 v8 v14 v13 v12 v3 v2 v1 v7 v6 v5 v4 v3 Figure 6 : T(7, 2, 2) Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 11 / 18

• Let Mi be a map on ni vertices. Let Ti = T(ri, si, ki) denote a (ri, si, ki)
• representation of Mi for i ∈ {1, 2}. Then, T1 ∼

= T2 ∀ r1 = r2, T1 ∼ = T2 ∀ s1 = s2, T(r1, s1, k1) ∼ = T(r1, s1, k2) for s1 = 2 and k2 ∈ {2, 3, . . . , r1 − 3} \ {k1, r1 − k1 − 1}, T(r1, s1, k1) ∼ = T(r1, s1, k2) for s1 ≥ 4 and k2 ∈ {0, 1, . . . , r1 − 1} \ {k1, r1 − k1 − s1

2 }, T(r1, s1, k1) ∼

= T(r1, s1, r1 − k1 − 1) for s1 = 2 and r1 ≥ 5, and T(r1, s1, k1) ∼ = T(r1, s1, r1 − s1

2 − k1) for s1 ≥ 4 and r1 ≥ 3.

Table : Maps of type {33, 42}

n Equivalence classes Length of cycles i(n) 10 T(5, 2, 2) (5, {10, 10}, 4) 1(10) 12 T(6, 2, 2), T(6, 2, 3) (6, {6, 4}, 4) 3(12) T(3, 4, 0), T(3, 4, 1) (3, {4, 12}, 4) T(3, 4, 2) (3, {12, 12}, 6) 14 T(7, 2, 2), T(7, 2, 4) (7, {14, 14}, 4) 2(14) T(7, 2, 3) (7, {14, 14}, 5) 16 T(8, 2, 2), T(8, 2, 5) (8, {8, 16}, 4) 5(16) T(8, 2, 3), T(8, 2, 4) (8, {16, 4}, 5) T(4, 4, 0), T(4, 4, 2) (4, {4, 8}, 4) T(4, 4, 1) (4, {16, 16}, 5) T(4, 4, 3) (4, {16, 16}, 7) 18 T(9, 2, 2), T(9, 2, 6) (9, {18, 6}, 4) 5(18) T(9, 2, 3), T(9, 2, 5) (9, {6, 18}, 5) T(9, 2, 4) (9, {18, 18}, 6) T(3, 6, 0) (3, {6, 6}, 6) T(3, 6, 1), T(3, 6, 2) (3, {18, 18}, 7) Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 12 / 18

Vertex-transitive maps and Non-vertex-transitive maps on the torus

The M is a map in Figure : T1 of type {3, 6, 3, 6} on the torus. Let G := C(u1, u2, u3, u4)∪C(u5, u6, u7, u8)∪C(v2, v10, v8, v16, v6, v14, v4, v12)∪ C(v3, v9, v5, v11, v7, v13, v1, v15). The G is a 2-regular graph which is disjoint union of four cycles. The action of Aut(M) is not vertex transitive

• n M as C(u1, u2, u3, u5) and C(v2, v10, v8, v16, v6, v14, v4, v12) are

components of G of different sizes. So, the map T1 is not vertex-transitive.

v1 v2 v3 v4 v5 v6 v7 v8 v1 u1 u2 u3 u4 u1 v9 v10 v11 v12 v13 v14 v15 v16 v9 u5 u6 u7 u8 u5 v7 v8 v1 v2 v3 v4 v5 v6 v7 Figure : T1 Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 13 / 18

a21 b4 b5 b6 a18 a4 a3 a2 a1 a17 a22 b7 b8 b9 b10 a19 a8 a7 a6 a5 a23 b11 b15 b12b13 b14 b16 b16 b15 a20 a12 a16 a11 a10 a9 a13 a20 a24 a15 a14 a24 a13 c1 b1 b2 b3 b18 b22 c6 c5 c4 c3 b21 b21 b17 c2 b24 b23 c10 c9 c8 c7 c14 c13 c12 c1 c16 c11 c15 c2 b19 b20 b17 a3 a3 a6 a7 a10 a11 a15 a2 a14 a5 a4 a9 a8 a13 a12 a1 a16 c21 c17 c17 c22 c18 c23 c19 c24 c21 c20

T2

Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 14 / 18

u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 u11u12 u12 u1 v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v1 w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 w11w12 w12 w1 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x1 x12 v5 v6 v7 v8 v9 v10 v11 v12 v1 v2 v3 v4 v4

T3

v1 v2 v3 v4 v5 v6 v1 v7 v8 v9 v10 v11 v12 v7 v14 v14 v15 v16 v17 v18 v13 v13 v19 v20 v21 v22 v23 v24 v19 v4 v5 v6 v1 v2 v3 v3

T4

Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 15 / 18

w1 w2 w3 w4 w5 w6 w7 w8 w9 w1 u3 u4 u5 u6 u7 u2 u1 u8 u9 w9 u9 v2 v2 v3 v4 v5 v6 v7 v8 v9 v1 u3 u3 u4 u5 u6 u7 u8 u9 u1 u2 x2 x3 x4 x5 x6 x1 x9 x7 x8 x9 x1

T5

u1 u2 u3 u4 u5 u6 u7 u8 u9u10 u11 u12 u1 w2 w2 w3w4 w5 w6 w7 w8 w9w10w11 w12 w1 u12 v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v1 x1 x2 x3 x4 x5 x6 x7 x8 x9 x11 x11 x12 x1 w3 x11 x12 v11 v11 v12 v1 v2 v3 v4 v5 v6 v7 v8 v9 v10

T6

Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 16 / 18

Example of non-transitive maps

Proposition : ([7]) (A) There exists a non-transitive {3, 6, 3, 6}-toroidal maps on n vertices if n ≡ 3 mod(6) or n ≡ 3 mod(9). (B) There exists a non-transitive {3, 122}-toroidal maps on n vertices if n ≡ 6 mod(12) or n ≡ 6 mod(18). (C) There exists a non-transitive {34, 6}-toroidal maps on n vertices if n ≡ 0 mod(6) or n ≡ 6 mod(12). (D) There exists a non-transitive {4, 6, 12}-toroidal maps on n vertices if n ≡ 0 mod(12) or n ≡ 12 mod(24). (E) There exists a non-transitive {3, 4, 6, 4}-toroidal maps on n vertices if n ≡ 0 mod(6) or n ≡ 6 mod(12). (F) There exists a non-transitive {4, 82}-toroidal maps on n vertices if gcd(n, 4t + 8) = gcd(n, 4t + 4) for some t ∈ {0} ∪ N. Proposition : ([3, 4]) The maps types {36}, {44}, {63}, {33, 42}, {32, 4, 3, 4} are vertex-transitive on the torus.

Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 17 / 18

• A. Altshuler, Construction and enumeration of regular maps on the torus, Discrete
• Math. 4 (1973) 201-217.
• U. Brehm and W. K¨

uhnel, Equivelar maps on the torus, European J. Combin. 29 (2008) 1843-1861.

• B. Datta and D. Maity, Semi-regular maps of torus and Klein bottle. (preprint)
• B. Datta and A. K. Upadhyay, Degree-regular triangulations of torus and Klein
• bottle. Proc. Indian Acad. Sci. Math. Sci. 115 (2005), 279-307.
• B. Gr¨

unbaum and G. C. Shephard, Tilings and patterns, A Series of Books in the Mathematical Sciences, W. H. Freeman and Company, New York, (1989).

• W. Kurth, Enumeration of Platonic maps on the torus, Discrete Math. 61 (1986)

71-83.

• D. Maity and A. K. Upadhyay, Enumeration of Semi-Equivelar Maps on the Torus,

(arXiv)

• S. Negami, Uniqueness and faithfulness of embedding of toroidal graphs, Discrete
• Math. 44 (1983) 161-180.
• A. K. Tiwari and A. K. Upadhyay, An Enumeration of Semi-Equivelar Maps on

Torus and Klein bottle, Math. Slovaca (to appear).

Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 18 / 18

Thank You

Dipendu Maity (IISc Bangalore) Semi-equivelar maps on the torus October 3, 2016 18 / 18