Ratio Mathematica Volume 45, 2023 Modular Coloring and Switching in Some Planar Graphs Sanma. G. R1 P. Maya2 Abstract For a connected graph G, let c: V (G) →ℤk (k ≥ 2) be a vertex coloring of G. The color sum σ(v) of a vertex v of G is defined as the sum in ℤk of the colors of the vertices in N (v) that is (v) = ∑ c(u)u∈N(v) (mod k). The coloring c is called a modular k-coloring of G if 𝜎(x) ≠ 𝜎(y) in ℤk for all pairs of adjacent vertices x, y ∈ G. The modular chromatic number or simply the mc-number of G is the minimum k for which G has a modular k- coloring. A switching graph is an ordinary graph with switches. For many problems, switching graphs are a remarkable straight forward and natural model, but they have hardly been studied. A vertex switching Gv of a graph G is obtained by taking a vertex V of G, removing the entire edges incident with V and adding edges joining V to every vertex which are not adjacent to V in G. In this paper we determine the modular chromatic number of Wheel graph, Friendship graph and Gear graph after switching on certain vertices. Here, we first define switching of graphs. Next, we investigating several problems on finding the mc(G) after switching of graphs and provide their characterization in terms of complexity. Keywords. Modular coloring, Modular chromatic number, Switching, Wheel Graph, Friendship graph, Gear graph. MSC AMS Classification 2020: 05C153 1Assistant Professor, Department of Mathematics, Sree Narayana College, Sivagiri, Varkala, India. Phone No:9446069833 Email id: sanmagr@gmail.com. 2 Assistant Professor, Department of Mathematics, Sree Devi Kumari Women’s College, Kuzhithurai, Kanyakumari Dist, Tamil Nadu, India Phone No:9442588393. Email id: drmaya009@gmail.com 3 Received on July 22, 2022. Accepted on October 15, 2022. Published on January 30, 2023.doi: 10.23755/rm.v45i0.984. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY license agreement. 74 mailto:sanmagr@gmail.com Sanma. G. R, P. Maya 1. Introduction We are encouraged by the modular colorings and the modular chromatic number of different graphs, where the chromatic number is defined as the color sum of all the neighboring vertices in 𝕫k. At this point of view, to the curiosity for minimizing the modular chromatic number, determined to switching in certain vertices in some graphs. For a vertex v of a graph G, let N (v) denote the neighborhood of v (the set of adjacent vertices to vertex v). For a graph G without isolated vertices, let c: V (G) → 𝕫k (k ≥ 2) be a vertex coloring of G where adjacent vertices may be colored the same. The color sum 𝜎(v) of a vertex v of G is defined as the sum in 𝕫k of the colors of the vertices in N(v) ,that is 𝜎(v) =∑ 𝑐(𝑢)𝑢𝜖𝑁(𝑣) [1, 2, 3]. The coloring c is called a modular sum k-coloring or simply a modular k-coloring of G, if 𝜎(x) ≠ 𝜎(y) in 𝕫k for all pairs x, y of adjacent vertices of G. A coloring c is called modular coloring if c is a modular k-coloring for some integar k ≥ 2.The modular chromatic number mc(G) is the minimum k for which G has a modular k- coloring.This concept was introduced by Okamoto, Salehi and Zhang [4, 5, 6, 8]. In order to distinguish the vertices of a connected graph and to differntiate the adjacent vertices of a graph with the minimum number of colors, the concept of modular coloring was put forward by Okamoto, Salehi and Zhang [6]. A graph H is the switching of a graph G with respect to the vertex v of G if V (G) = V (H) and E(H) = (E(G)\{ uv : u ∈ NG(v) }) ∪ { uv : u ∈N’G(v) }[7,9]. The switching of G with respect to v is denoted Gs (v). The operation of creating Gs (v) is called switching on v in G. In other words, switching on a vertex v of a graph has the effect of removing all edges incident with the vertex v and joining the vertex v to all vertices to which it was formerly non-adjacent. Here, we first define switching of graphs. Next, we investigating several problems on finding the mc (G) after switching of graphs and provide their characterization in terms of complexity. In this paper we find the modular chromatic number of wheel graph, friendship graph and gear graph after switching on certain vertices at different levels. 2. Modular coloring of wheel graph after switching The switching of a vertex in a wheel having n vertices is denoted by Ws (n). In Wheel switching is not possible in W (3) since it is a complete graph. Switching in a wheel is obtained in two ways. They are 1)The switching at the vertex u ∈ ℓ0. 2)The switching at the vertex vi ∈ ℓ1.be the vertices Let u ∈ ℓ0 be the central vertex and v1, v2, …, vn ∈ ℓ1be the vertices which are adjacent to u ∈ ℓ0. The switching at u ∈ ℓ0 makes the graph W(n) is a cycle having n vertices with a central vertex u ∈ ℓ0. 75 Modular Coloring and Switching In Some Planar Graphs Theorem 2.1 The modular coloring of a graph obtained after the switching of a vertex vi ∈ ℓ1 is Ws (n) = 3 for n = 4k, 4k+1, [4k+2; k > 1]; Ws (6) = 4; Ws (n) = 4 for n = 4k+3, k ≥ 1. Proof: For a wheel W (n), let the vertex u ∈ ℓ0 ,vi ∈ ℓ1 for I = 1, 2, …., n be the vertices. Case (i) mc [Ws (4)] = 3. Let v1, v2, v3, v4 ∈ ℓ1be the vertices at level ℓ=1.Switching is taken forv1 ∈ ℓ1. Consider the modular coloring c(v):v[Ws(4)]→ 𝕫3 defined by c(v)={ 0 for v1 ∈ ℓ1, u ∈ ℓ0 1 otherwise then 𝜎(v)={ 0 𝑓𝑜𝑟 𝑢 ∈ ℓ0 2 𝑓𝑜𝑟 𝑣3 ∈ ℓ1 1 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in Ws(4). ∴ mc [ws (4)]=3. Hence the proof. Case (ii) mc [Ws (5)] = 3. Let v1, v2, v3, v4, v5∈ ℓ1be the vertices at level ℓ=1.Switching is taken for v1 ∈ ℓ1. Consider the modular coloring c (v):v[Ws(5)]→ 𝕫3 defined byc(v)={ 1 for v3 ∈ ℓ1 2 for v4 ∈ ℓ1 0 otherwise then σ(v) ={ 0 for u ∈ ℓ0, v1 ∈ ℓ1 1 for v2,v4 ∈ ℓ1 2 otherwise Here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀x,y of adjacent vertices in Ws(5). ∴ mc [ws (5)]=3.Hence the proof. Case (iii)mc[Ws(6)] = 4.Let v1,v2,v3,v4,v5,v6∈ ℓ1be the vertices at level ℓ=1.Switching is taken for v1 ∈ ℓ1 Consider the modular coloring c(v):v[Ws(6)]→ 𝕫3 defined by c(v) = { 1 for v3 ∈ ℓ1 2 for v4 ∈ ℓ1 0 otherwise Then σ (v) = { 3 for u ∈ ℓ0, v1 ∈ ℓ1 1 for v2,v4 ∈ ℓ1 2 for v3,,v5 ∈ ℓ1 0 otherwise Here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in Ws(6). ∴ mc [ws (6)] = 3. Hence the proof. Case (iv) mc [Ws (4k+3)] = 4. Let u∈ ℓ0 be the central vertex. Let v1,v2,…vi-1,vi,vi+1,….,v4k+3∈ ℓ1be the vertices at level ℓ =1.Switching is taken for vi ∈ ℓ1 . After switching vi is adjacent to the vertices vi+2,vi+3,…v4k+3,v1,v2,…,vi-2 respectively and not adjacent to the vertices vi-1 and vi+1. The 4k vertices which are adjacent to vi is renamed as R1, R2, …., R4k. 76 Sanma. G. R, P. Maya Subcase (i) mc [Ws (4k+3)] = 4 for k = 1 + 4j, j = 0, 1, 2, …. [Eg: Ws (7), Ws (23), Ws (39), …. Consider the modular coloring c(v):v[Ws(4k+3)]→ 𝕫4 defined by c(v)={ 2 if R4k ∈ ℓ1 1 if R1+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere then σ(v)={ 3 for u ∈ ℓ0, vi ∈ ℓ1 2 for vi−1,R4k−1 ∈ ℓ1 1 for vi+1,,R2j ∈ ℓ1, for j = 1,2, … (2k − 1) 0 otherwise Here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in Ws(4k+3). ∴ mc[ws(4k+3)]=4 for k=1+4j ,j=0,1,2,….Hence the proof. Eg: Figure1. Switching with modular coloring in Ws(7) Subcase(ii) mc[Ws(4k+3)]=4 for k=2+4j ,j=0,1,2,….[Ws(11),Ws(27), Ws(43),…] Consider the modular coloring c(v):v[Ws(4k+3)]→ 𝕫4 defined by c(v)={ 2 if R4k ∈ ℓ1 1 if u ∈ ℓ0, R1+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere then σ(v)={ 0 for u ∈ ℓ0, vi ∈ ℓ1 3 for R4k−1 ∈ ℓ1 2 for vi+1,,𝑣𝑖−1, 𝑅2𝑗 ∈ ℓ1, for j = 1,2, … (2k − 1) 1 otherwise Here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in Ws(4k+3). ∴ mc[ws(4k+3)]=4 for k=2+4j ,j=0,1,2,…. Hence the proof. Eg: 77 Modular Coloring and Switching In Some Planar Graphs Figure 2. Switching with modular coloring in Ws(11) Subcase(iii) mc[Ws(4k+3)]=4 for k=3+4j ,j=0,1,2,….[Eg: Ws(15), Ws(31),Ws(47),…] Consider the modular coloring c(v):v[Ws(4k+3)]→ 𝕫4 defined by c(v)={ 2 if u ∈ ℓ0, R4k ∈ ℓ1 1 if R1+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere then σ(v)={ 0 for 𝑣𝑖−1 , R4k−1 ∈ ℓ1 1 for u ∈ ℓ0, 𝑣𝑖 ∈ ℓ1 3 for vi+1,,𝑅2𝑗 ∈ ℓ1, for j = 1,2, … (2k − 1) 2 otherwise Here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in Ws(4k+3). ∴ mc[ws(4k+3)]=4 for k=3+4j ,j=0,1,2,….Hence the proof. Eg: Figure 3. Switching with modular coloring in Ws(15) Subcase(iv) mc[Ws(4k+3)]=4 for k=4+4j ,j=0,1,2,….[Ws(19),Ws(35),Ws(51),…] 78 Sanma. G. R, P. Maya Consider the modular coloring c(v):v[Ws(4k+3)]→ 𝕫4 defined by c(v)={ 3 if u ∈ ℓ0 1 if R1+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 2 if R4k ∈ ℓ1 0 elsewhere then σ(v)={ 1 for vi−1 , R4k−1 ∈ ℓ1 2 for u ∈ ℓ0, vi ∈ ℓ1 0 for vi+1,,R2j ∈ ℓ1, for j = 1,2, … (2k − 1) 3 otherwise Here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in Ws(4k+3). ∴ mc[ws(4k+3)]=4 for k=4+4j ,j=0,1,2,….Hence the proof. Eg: Figure 4. Switching with modular coloring in Ws(19) Case(v)mc[Ws(4k)] = mc[Ws(4k+1)] = mc[Ws(4k+2)]=3. Subcase(i) mc[Ws(4k)] = 3 for k=2+3j,j=0,1,2….[Ws(8),Ws(20),Ws(32),…] After switching vi is adjacent to the vertices vi+2,vi+3,…v4k,v1,v2,…,vi-2 respectively and not adjacent to the vertices vi-1 and vi+1. Let the 4k-3 vertices which are adjacent to vi is renamed as R1,R2,…..,R4k-3 respectively. Consider the modular coloring c(v):v[Ws(4k)]→ 𝕫3 defined by c(v)={ 1 if R1+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere then σ(v)={ 2 for u ∈ ℓ0, vi ∈ ℓ1 1 for vi+1,,vi−1, R2j ∈ ℓ1, for j = 1,2, … . ,2(k − 1) 0 otherwise Here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in Ws(4k). ∴ mc[ws(4k)]=3 for k=2+3j ,j=0,1,2,….Hence the proof. Eg: 79 Modular Coloring and Switching In Some Planar Graphs Figure 5. Switching with modular coloring in Ws(8) Subcase(ii) mc[Ws(4k)] = 3 for k=3+3j,j=0,1,2….[Ws(12),Ws(24),Ws(36),..] After switching vi is adjacent to the vertices vi+2,vi+3,…v4k,v1,v2,…,vi-2 respectively and not adjacent to the vertices vi-1 and vi+1. Let the 4k-3 vertices which are adjacent to vi is renamed as R1, R2, ….., R4k-3 respectively. Consider the modular coloring c(v):v[Ws(4k)]→ 𝕫3 defined by c(v)={ 1 if u ∈ ℓ0, R1+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere then σ(v)={ 0 for u ∈ ℓ0, vi ∈ ℓ1 1 R1+2j ∈ ℓ1, for j = 0,1,2, … . , (k + 1) 2 otherwise Here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in Ws(4k). ∴ mc[ws(4k)]=3 for k=3+3j ,j=0,1,2,….Hence the proof. Eg: Figure 6. Switching with modular coloring in Ws(12) Subcase(iii) mc[Ws(4k)] = 3 for k=4+3j,j=0,1,2….[Ws(16),Ws(28),Ws(40),…] After switching vi is adjacent to the vertices vi+2,vi+3,…v4k,v1,v2,…,vi-2 respectively and not adjacent to the vertices vi-1 and vi+1. Let the 4k-3 vertices which are adjacent to vi is renamed as R1, R2, ….., R4k-3 respectively. 80 Sanma. G. R, P. Maya Consider the modular coloring c(v):v[Ws(4k)]→ 𝕫3 defined by c(v)={ 2 if u ∈ ℓ0 1 forR1+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere then σ(v)={ 1 for u ∈ ℓ0, vi ∈ ℓ1 2 R1+2j ∈ ℓ1, for j = 0,1,2, … . ,2(k − 1) 0 otherwise Here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in Ws(4k). ∴ mc[ws(4k)]=3 for k=4+3j ,j=0,1,2,…. Hence the proof. Eg: Figure7. Switching with modular coloring in Ws(16) Subcase(iv) mc[Ws(4k+1)] = 3 for k=2+3j,j=0,1,2….[Ws(9),Ws(21),Ws(33),…] After switching vi is adjacent to the vertices vi+2,vi+3,…v4k+1,v1,v2,…,vi-2 respectively and not adjacent to the vertices vi-1 and vi+1. Let the 4k-2 vertices which are adjacent to vi is renamed as R1, R2… R4k-2 respectively. Consider the modular coloring c(v): v[Ws(4k+1)]→ 𝕫3 defined by c(v) = { 1 forR1+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere Then σ(v) = { 2 for u ∈ ℓ0, vi ∈ ℓ1 1 for vi+1 , R2j ∈ ℓ1, for j = 1,2, … . , (2k − 1) 0 otherwise Here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x, y of adjacent vertices in Ws (4k+1). ∴ mc [ws (4k+1)] = 3 for k = 2+3j, j = 0, 1, 2, …. Hence the proof. 81 Modular Coloring and Switching In Some Planar Graphs Eg: Figure 8. Switching with modular coloring in Ws(9) Subcase (v) mc[Ws (4k+1)] = 3 for k = 3+3j, j = 0, 1, 2…. [Ws (13), Ws (25), Ws (37), …] After switching vi is adjacent to the vertices vi+2, vi+3, …v4k+1, v1, v2,…,vi-2 respectively and not adjacent to the vertices vi-1 and vi+1. Let the 4k-2 vertices which are adjacent to vi is renamed as R1, R2, ….., R4k-2 respectively. Consider the modular coloring c(v):v[Ws(4k+1)]→ 𝕫3 defined by c(v)={ 1 for u ∈ ℓ0 ; R1+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere Then σ(v) = { 0 for u ∈ ℓ0, vi ∈ ℓ1 2 for vi+1 , R2j ∈ ℓ1, for j = 1,2, … . , (2k − 1) 1 otherwise Here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in Ws(4k+1). ∴ mc[ws(4k+1)] = 3 for k = 3+3j, j = 0, 1, 2, …. Hence the proof. Eg: Figure 9. Switching with modular coloring in Ws(13) Subcase (vi) mc [Ws(4k+1)] = 3 for k = 4+3j, j = 0, 1, 2…. [Ws (17), Ws (29), Ws (41), …] 82 Sanma. G. R, P. Maya After switching vi is adjacent to the vertices vi+2,vi+3,…v4k+1,v1,v2,…,vi-2 respectively and not adjacent to the vertices vi-1 and vi+1. Let the 4k-2 vertices which are adjacent to vi is renamed as R1, R2, ….., R4k-2 respectively. Consider the modular coloring c(v):v[Ws(4k+1)]→ 𝕫3 defined by c(v)={ 2 for u ∈ ℓ0 1 for R1+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere then σ(v)={ 1 for u ∈ ℓ0, vi ∈ ℓ1 0 for 𝑣𝑖+1 , R2j ∈ ℓ1, for j = 1,2, … . , (2k − 1) 2 otherwise Here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in Ws(4k+1). ∴ mc[ws(4k+1)]=3 for k=4+3j ,j=0,1,2,….Hence the proof. Eg: Figure 10. Switching with modular coloring in Ws(17) Subcase(vii) mc[Ws(4k+2)] = 3 for k=2+3j,j=0,1,2….[Ws(10),Ws(22),Ws(34),…] After switching vi is adjacent to the vertices vi+2, vi+3, …v4k+2, v1, v2, …, vi-2 respectively and not adjacent to the vertices vi-1 and vi+1. Let the 4k-2 vertices which are adjacent to vi is renamed as R1, R2, ….., R4k-1 respectively. Consider the modular coloring c(v):v[Ws(4k+2)]→ 𝕫3 defined by c(v)={ 1 for R2+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere then σ(v)={ 2 for u ∈ ℓ0, vi ∈ ℓ1 1 for R1+2j ∈ ℓ1, for j = 0,1,2, … . , (2k − 1) 0 otherwise Here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in Ws(4k+2). ∴ mc [ws (4k+2)] = 3 for k = 2+3j, j = 0, 1, 2, …. Hence the proof. 83 Modular Coloring and Switching In Some Planar Graphs Eg: Figure 11. Switching with modular coloring in Ws(10) Subcase(viii) mc[Ws(4k+2)] = 3 for k=3+3j,j=0,1,2….[Ws(14),Ws(26),Ws(38),…] After switching vi is adjacent to the vertices vi+2,vi+3,…v4k+2,v1,v2,…,vi-2 respectively and not adjacent to the vertices vi-1 and vi+1. Let the 4k-2 vertices which are adjacent to vi is renamed as R1,R2,…..,R4k-1 respectively. Consider the modular coloring c(v):v[Ws(4k+2)]→ 𝕫3 defined by c(v)={ 1 for u ∈ ℓ0 , R2+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere then σ(v)={ 0 for u ∈ ℓ0, vi ∈ ℓ1 2 for R1+2j ∈ ℓ1, for j = 0,1,2, … . , (2k − 1) 1 otherwise Here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in Ws(4k+2). ∴ mc[ws(4k+2)]=3 for k=3+3j ,j=0,1,2,….Hence the proof. Eg: Figure 12. Switching with modular coloring in Ws(14) Subcase(ix)mc [Ws(4k+2)] = 3 for k=4+3j,j=0,1,2….[Ws(18),Ws(30),Ws(42),…] 84 Sanma. G. R, P. Maya After switching vi is adjacent to the vertices vi+2,vi+3,…v4k+2,v1,v2,…,vi-2 respectively and not adjacent to the vertices vi-1 and vi+1. Let the 4k-2 vertices which are adjacent to vi is renamed as R1,R2,…..,R4k-1 respectively. Consider the modular coloring c(v):v[Ws(4k+2)]→ 𝕫3 defined by c(v)={ 2 for u ∈ ℓ0 1 𝑓𝑜𝑟 R2+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere then σ(v)={ 1 for u ∈ ℓ0, vi ∈ ℓ1 0 for R1+2j ∈ ℓ1, for j = 0,1,2, … . , (2k − 1) 2 otherwise Here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in Ws(4k+2). ∴ mc[ws(4k+2)]=3 for k=4+3j ,j=0,1,2,….Hence the proof. Eg: Figure 13. Switching with modular coloring in Ws(18) 3. Modular colorings after switching on Friendship graph Let u ∈ ℓ0 be the center v1, v2, v3, …. v2n be the vertices in ℓ1 where each of 2 consecutive vertices forms an edge for the respective cycles since a friendship graph is constructed by joining n copies of the cycle C3 with a common vertex. The vertices in ℓ1is taken in the clockwise direction. Modular coloring after switching of a friendship graph with n vertices is denoted by mc[FSs(n)].Here switching can be taken only for vi∈ ℓ1 for any i=1,2,… 2n.We cannot form a switching with u ∈ ℓ0 since it is adjacent to all vertices of ℓ1. Theorem 3.1. The modular coloring of a friendship graph after switching a vertex in ℓ1 then (i)mc[FSs(2)]=3. (ii) mc[FSs(n)]=4;n≥ 3. Proof: Case (i) mc[FSs(2)] = 3. 85 Modular Coloring and Switching In Some Planar Graphs Switching is taken for v1∈ ℓ1.Therefore v1 is adjacent to v3 and v4 after switching in FSs(2). Consider a modular coloring c(v):v[FSs(2)] → 𝕫3 defined by C(v)= { 2 for u ∈ ℓ0 1 for v4 ∈ ℓ1 0 elsewhere then σ(v)={ 1 for u ∈ ℓ0, v1 ∈ ℓ1 2 for v4, v2 ∈ ℓ1 0 otherwise Here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀x,y of adjacent vertices in FSs(2) ∴mc[FSs(2)]=3. Hence the proof. Case(ii) mc[FSs(n)]=4;n≥3. Subcase (i) mc[FSs(n)]=4;n≥3 for n is odd. Switching is taken for v1∈ ℓ1.Therefore v1 is adjacent to v3 ,v4 ,….v2n after switching in FSs(n). Consider a modular coloring c(v):v[FSs(n)] → 𝕫4 defined by C(v)= { 3 for u ∈ ℓ0 2 for v2j ∈ ℓ1 ; j = 1,2, … , n 0 elsewhere then σ(v)={ 2 for u ∈ ℓ0 3 v2j ∈ ℓ1; 𝑗 = 1,2, … . , 𝑛 0 for v1 ∈ ℓ1 1 otherwise Here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in FSs(n) ∴ mc[FSs(n]=4 for n≥3 for n is odd. Hence the proof. Eg: Figure 14. Switching with modular coloring in FSs(5) 86 Sanma. G. R, P. Maya Subcase (ii).mc[FSs(n)]=4;n≥3 for n is even. Switching is taken for v1∈ ℓ1.Therefore v1 is adjacent to v3 ,v4 ,….v2n after switching in FSs(n). Consider a modular coloring c(v):v[FSs(n)] → 𝕫4 defined by C(v)= { 3 for u ∈ ℓ0 2 for v2j ∈ ℓ1 ; j = 1,2, … , n 0 elsewhere then σ(v)={ 0 for u ∈ ℓ0 3 v2j ∈ ℓ1; 𝑗 = 1,2, … . , 𝑛 2 for v1 ∈ ℓ1 1 otherwise Here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in FSs(n) ∴ mc[FSs(n]=4 for n≥3 for n is even. Hence the proof. Eg: Figure 15. Switching with modular coloring in FSs(4) 4.Modular colorings after switching on Gear graph. Let u ∈ ℓ0 be the center of a gear graph.Let v1,v3,v5,….v2n-1 be the vertices in ℓ1 are adjacent to u ∈ ℓ0 and v2,v4,v6,….v2n be the vertices in ℓ1 .The switching in G(n)is denoted by Gs(n).Switching in gear graph is obtained in two ways. That is (i)switching of the vertex u ∈ ℓ0 and (ii)switching of a vertex vi ∈ ℓ1; 𝑖 = 1,2, … . ,2𝑛. (i)By switching of the vertex u ∈ ℓ0 in a gear graph G(n) result in another gear graph G’(n) in which vertices in ℓ1which are not adjacent with u ∈ ℓ0 in G(n) become adjacent with G’(n).Therefore Gs(n)=G(n)=G’(n).Hence mc[Gs(n)]=mc[G(n)]=mc[G’(n)]. (ii) switching of a vertex vi ∈ ℓ1; 𝑖 = 1,2, … . ,2𝑛 .Here specifying the vertex vi ∈ ℓ1 which are adjacent with u ∈ ℓ0 is taken for switching.In general take vi as v1. 87 Modular Coloring and Switching In Some Planar Graphs Theorem 4.1 The modular coloring of the graph obtained after the switching of a vertex in Gear graph in ℓ1 (which are adjacent with u ∈ ℓ0).ie (i) mc[Gs(2)]=2.(ii) mc[Gs(n)]=3;n>2. Proof. Case (i) mc[Gs(2)]=2. Switching is taken for v1∈ ℓ1in G (2).Therefore v1 is adjacent to v3 after switching in Gs (n). Consider a modular coloring c (v):v[Gs(2)] → 𝕫2 defined by C(v)= { 1 for v3 ∈ ℓ1 0 elsewhere Then σ(v)={ 0 v3 ∈ ℓ1 1 otherwise Here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in Gs(2) ∴mc [Gs(2]=2. Hence the proof. Case (ii) mc[Gs(n)]=3 for n>2. Switching is taken for v1∈ ℓ1 in G(n).Therefore v1 is adjacent to v3 ,v4,….v2n-1 after switching in Gs(n) and not adjacent to the remaining vertices in Gs(n). Consider a modular coloring c(v):v[Gs(n)] → 𝕫3 defined by C(v)= { 1 for u ∈ ℓ0; v1 ∈ ℓ1 0 elsewhere then σ(v)={ 2 for v3+2j ∈ ℓ1 ; 𝑗 = 0,1,2, … . (𝑛 − 2). 1 for v4+2j ∈ ℓ1 ; 𝑗 = 0,1,2, … . (𝑛 − 3) 0 otherwise Here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in Gs(n) ∴ mc[Gs(n)]=3 for n>2. Hence the proof. Eg: Figure 16. Switching with modular coloring in Gs(6). 88 Sanma. G. R, P. Maya 5. Conclusions In a Wheel Graph the modular coloring of a graph obtained after the switching of a vertex vi ∈ ℓ1 is Ws(n)=3 for n=4k,4k+1,[4k+2;k>1]; Ws(6)=4; Ws(n)=4 for n=4k+3,k≥1.The labeling is quite similar to one other and differs according to the change in number of vertices. Also in a Friendship Graph the modular coloring after switching a vertex in ℓ1 then (i)mc[FSs(2)]=3. (ii) mc[FSs(n)]=4;n ≥ 3. Similarly in Gear Graph the modular coloring obtained after the switching of a vertex in ℓ1 (which are adjacent with u ∈ ℓ0).ie (i) mc[Gs(2)]=2.(ii) mc[Gs(n)]=3;n>2. Altogether It is explicitly clear that after switching in different levels of the graphs, the modular chromatic number varies in between two to four. We cannot expect a higher level of modular chromatic number after switching in vertices at different levels. 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