Rainbow Vertex Coloring for Line, Middle, Central, Total Graph of Comb Graph

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1 Idia Joural of Sciece ad Techology, Vol 9(S, DOI: /ijst/206/v9iS/97463, December 206 ISSN (Prit : ISSN (Olie : Raibow Vertex Colorig for Lie, Middle, Cetral, Total Graph of Comb Graph C. S. Hariramkumar * ad N. Parvathi Faculty of Egieerig ad Techology, Departmet of Mathematics, SRM Uiversity, Kattakulathur , Tamil Nadu, Idia; hariramkumar8@gmail.com, Parvathi.@ktr.srmuiv.ac.i Abstract Objectives: To fid the raibow vertex coectio umber for Lie, Cetral, Middle ad Total graphs of Comb Graph. Methods/Statistical Aalysis: The methods to fid the raibow vertex coectio umber of ay graph G is quite differet from other colorig problems. Fidigs: The raibow vertex coectio umber for lie graph ad middle graph of comb graph is raibow vertex coectio umber for Cetral graph of Comb graph is 2 ad Total graph of Comb graph is if =7, ; is 3 - if =4k+, k. While fidig the achromatic umber for ay graph G, otig that o two adjacet vertices receives same color but i the case of raibow vertex colorig we ca assig same color to the adjacet vertices. Applicatio/Improvemets: The applicatios of raibow vertex coectio umber are same as raibow coectio umber. Keywords: Cetral ad Total Graph of Comb Graph, Lie, Middle, Raibow Vertex Colorig, Raibow Vertex Coectio Number. Itroductio I make-kow the Raibow Coectio i which the ideas are implemeted i may applicatios. Raibow coectio expresses ideas as takig distict passwords i the itermediate agecies coectig betwee ay two agecies. The Cocept Raibow Coectio implemeted o Iformatio security purposes betwee ay two agecies. This applicatio is to share secured iformatio i ay path betwee ay two agecies, provided the iformatio caot view by the third party, which leads to security purposes. This applicatio are modeled i graph theory as edges ca be treated as passwords, adjacet edges are assig same color but i the path wise, o two edges have same color coectig betwee ay two vertices. All graphs are fiite, simple ad udirected. Ay otatio or termiology follows from the 2 book. The existece of raibow vertex coectio umber was brought up by the article 3. The 4 defiitios of Lie Graph ad Total Graph of ay graph are take. A -regular caterpillar is called a Comb. 5 9 Shows the other refereces. I Graph Theory, the coectivity is the fudametal graph-theoretic subject i combiatorial, algorithmic sese. There are may cocepts existed due to applicatios raised ad durable for the servig the purpose. Amog them, the ew ad fabricatig cocept is raibow coectio, which provides stregth to coectivity. 2. Mai Results. Lie Graph of Comb Graph Theorem 4. Let G = L(CG ] where diam (G 2, the rvc(g = Proof : Let x єv(g, x is peripheral vertex i G ad diam (G 2. Let T i (x = {v: d(x,v=i} where iє[,diam(g]. Let U T i (x (i is eve =V ad U T i (x (i is odd =. ad V 3 = {x} where x is peripheral vertex i G. Case : diam (G is eve of atleast 2 *Author for correspodece

2 Raibow Vertex Colorig for Lie, Middle, Cetral, Total Graph of Comb Graph For = 3, Here diam (G = 2 the rvc (G =. Now we assume diam (G 4 Defie a colorig c:v(g [, ] to the vertices of G as follows, sice V = U T i (x( I diam(g, where i is eve, = U T i (x ( i diam(g- is odd. Assig color to the vertices u 2k- (kєn ad also tou 2, the assigig remaiig - distict colors to the vertices u 2k (k 2 as show i the Figure. Let P,P 2 P deotes paths, ot ecessarily distict. Let x, x 2,..x be the iteral vertices i the path P, Where the vertices are colored with colors,,2.,ad c(x c(y where x ad y are ot iitial ad termial vertices i the path P. The the path P is raibow vertex coected. Now we assume that d (u,u - < 2k,the vertices are u,u 2,u 4..u - are colored with colors,,2. The the path P2 : u,u 2,u 4..u - is raibow vertex-coected of legth less tha 2k.. Hece rvc(g.. Suppose that if we take rvc(g< colors, the ay two iteral vertices x ad y i the path of legth 2k where c(x = c(y, which is ot a raibow path. Therefore rvc(g. If we have take rvc(g> colors which is also a raibow vertex coected with maximum colors chose.thusrvc(g = Case 2: diam (G is odd where diam (G 3 SiceV =U T i (x where 2 i diam (G-, i is eve ad = U T j (x where j diam (G, j is odd Let c(u 2k- =, kєn ad also to u 2 ad assigig remaiig - distict colors to the vertices u 2k (k 2. Let P, P 2 P deotes paths, ot ecessarily distict. Let x, x 2,..x be the iteral vertices i the path P, oe ca easily fid that c(x c(y where x ad y are ot iitial ad termial vertices i the path P. The path P is raibow vertex coected. Now we assume that d (u,u - <diam (G, the vertices are colored with colors,,2.,, Hece rvc (G. Suppose that we have a vertex colorig for G with fewer tha rvc(g colors, ay two iteral vertices x,y i the path of legth diam (G have same color, the we do ot have a raibow path. Therefore rvc (G. O the other had, if we take rvc (G > colors which is also a raibow vertex coected with maximum colors chose. Thus rvc (G = Figure. Lie graph of comb graph. 3. Raibow Vertex Coectio Number for Middle Graph of Comb Graph Theorem 5.: Let G = M[CG ] of eve order, where 2 the rvc(g = Proof :.Let diam (G =k 4.Let u,v Є V(G be such that d G (u,v = d where d G (u,v represets distace betwee u ad v. Case : = 8k-, k Defie colorig to the vertex set V(G as follows :c(u i = c(v i =, i є [,] ad assigig the remaiig - distict colors to the vertices u i where i є [2,4k-] ad c(u i =. Without loss of geerality u =v ad v=v ad let d G (u,v=k. Let P,.P be paths, ot ecessarily distict, Amog the paths P i ( i,let P l be the diameter path whose iteral vertices have distict colors.thus rvc(g By takig rvc(g < colors. Defie a vertex col- orig c : V(G [, -] as follows : c(v i = c(u i = for ( i 2k ad c(u = c(u = ad c(u 2k = 2k, k ad c(u i = i, where i,4k-,i 3, i is odd. Sice P l is the diameter path, where c( u = c(u, the the path is ot raibow path. Thus rvc(g 2 Vol 9 (S December Idia Joural of Sciece ad Techology

3 C. S. Hariramkumar ad N. Parvathi By takig rvc (G > colors, the there will be raibow path with maximum colors chose,. Thus rvc (G = Case 2:- = 8k+3, k Let V ={ v i /i ε[,2k+]}, /i ε[,2k+} ad V 3 = { u i /i ε[,4k+]}. Defie a vertex colorig c:v(g [, ] as follows. Let c(v i =c(u i =for i ε[,2k+]assigig the remaiig - distict colors to the vertices u i whereiє [,4k+] ad by lettig c(u 2k- =2k-, k ad c(u 2k = 2k, k є N as show i the Figure 2. Figure 2. Middle graph of comb graph. Without loss of geerality assume u = v ad v = v ad let d G (u,v = k. Let P,.P be paths, ot ecessarily distict, Amog the paths P i ( i,let P l be the diameter path whose iteral vertices have distict colors.the there exists a raibow path. Thus rvc (G Suppose by takig rvc (G < colors, the we have to show that there will be o raibow vertex coected graph. Defie a vertex colorig c : V(G [, -] as follows c(v i = c(u i = for ( i 2k ad c(u = c(u = ad c(u = 2k, k ad c(u 2k = i, where i,4k+,i 3, i i is odd. Sice P l is the diameter path, c(u = c(u where u ad u are ot iitial ad termial vertices, Thus rvc (G. Suppose if we takig rvc (G >, the there will be raibow path with maximum colors rvc (G =. chose,. Thus 4. Raibow Vertex Coectio Number for Cetral Graph of Comb Graph Theorem 6. Let G be a Cetral Graph of Comb Graph with diameter d = 3, the rvc (G = 2. Proof: For = 7, it is obvious that rvc (G = 2. For, we distiguish the followig cases accordig to maximum degree Δ (G. Case : Δ (G = 4k+, k. Now we assume that =8k+3, k. Let G be a graph with vertex set V(G = V U where V ={v i / i 2k+}, / i 2k+} ad V 3 = {u i / i 4k+}. Defie a vertex colorig c:v(g [,2] as follows. Assig color to the vertex v ad assig color 2 to the remaiig vertices of V,c(u i = for every i( i 2k+ ad c(u i = for ( i 4k+ as show i the Figure 3. Every path is raibow vertex coected with this colorig. Thus rvc(g 2. To show that rvc (G 2. First we show that rvc (G. Suppose if we are colorig the vertices with oly oe color as follows, c(u i = c(v i = for i 2k+ u ad c(u i = u 4k + for i 4k+. Let P be a path u - v v 3 - u 4k +, sice c(v =c(v 3 =, there exists o raibow path P. Thus rvc (G.Now we show that rvc (G 2,Suppose c:v (G [i] where i 3,the every path is raibow vertex coected with maximum colors Thus rvc (G = 2. Figure 3. Total graph of comb graph. Case 2: Δ (G=4k+3, k. Now we assume that = 8k+7, k. Let G be a graph with vertex set V (G = V U where V ={v i / i 2k+2}, Vol 9 (S December Idia Joural of Sciece ad Techology 3

4 Raibow Vertex Colorig for Lie, Middle, Cetral, Total Graph of Comb Graph / i 2k+2} ad V 3 = {u i / i 4k+3}. Defie a vertex colorig c:v(g [,2] as follows. c(v =c(u i =for every i ( i 2k+2, c(u i = for i 4k+3 ad assigig color 2 to the remaiig vertices of V.Every path is raibow vertex coected with this colorig. Thus rvc (G 2. To show that rvc (G 2. The same argumet follows from case. 7. Raibow Vertex Coectio Number for Total Graph of Comb Graph Theorem 7. : Let G = T[CG ] of eve order where 2,the rvc(g = if =7, ad rvc(g = 3 - if 3 =4k+, k є N Proof: Let G be a graph with vertex set V (G = V U where V ={v i : i }, : i } ad V 3 = {u i : i }.We distiguish the proof accordig to the order of graph G. Case : =7, For = 7,. Defie a vertex colorig c:v(g [, ] as follows.c(v i = for i 3 c(u i = for i 3 ad c(u i = i for i.the there exists a raibow path coectig betwee ay two vertices i G,so c is a raibow vertex colorig. Thus rvc(g. 3 Now, we show that rvc(g.if we have vertex colorig fewer tha 3 3 colors, the the two vertices i the iteral vertices have same color. Let P, P are paths, ot ecessarily distict. let P l be a path of legth exactlydiam (G. let u,v ε V(G, the some two vertices u ad v i the iteral vertices have same color, the there will be o raibow path. Hece by takig rvc(g< 3 colors, which cotradicts the defiitio of raibow vertex coected. Hece rvc(g 3. If we have vertex colorig more tha colors the 3 there exists a raibow path coectig betwee every pair of vertices with maximum colors, which cotradicts the defiitio of raibow vertex coectio umber.thus rvc(g = will be miimum compared to rvc(g >. 3 3 Case 2: Now, we assume that = 4k+, k Let us cosider the two subcases to show that rvc (G = Subcase 2.: For = 8k+7, where k Let G be a graph with vertex set V(G = V U where V = {v i ; i 2k+2} ad ; i 2k+2} ad V 3 ; i 4k+3}. Defie a vertex colorig c : V(G [, ] as follows. c(v i = for ( i 2k+2 ad c(u i = for ( i 4k+3 ad assigig the remaiig colors to the vertices of V(G-V -V 3 as follows, c(u i =i for ( i 2k+2 as show i the Figure 4 Let P,.P be a paths, ot ecessarily distict.amog the paths P i ( i,let P l be the path havig distace d G (u,v=diam(g, whose iteral vertices have distict colors. The every path is raibow Vertex coected, obviously the graph is raibow vertex coected graph. Thus rvc(g.if we colorig the vertices fewer tha colors. Suppose if we colorig the vertices with 2 colors, the we defie the vertex colorig c 2 :V(G 2 as follows. Assigig the color to the vertices of degree exactly 2 ad c 2 (u i = for i, ad c 2 (u 2k- =2k-, k εn, c 2 (u i =i, i -,where i, are eve ad c(u i = if i=. Let P l be the path havig distace d G (u,v=diam(g where two vertices say u ad u exists i the iteral vertices i the path P l have same color, the the path is ot a raibow path, the the graph is ot raibow vertex coected graph. Thus if we have fewer tha colors, the the 3 graph is ot raibow vertex coected graph. 4 Vol 9 (S December Idia Joural of Sciece ad Techology

5 C. S. Hariramkumar ad N. Parvathi Thus rvc(g 3.If we have vertex colorig c more tha 3 colors, the there will be raibow vertex colorig with maximum colors chose which is cotradictio to the defiitio of raibow vertex coectio umber. Thus rvc(g = 3 colors will be miimum, which is ot cotradictio to the defiitio of raibow vertex coectio umber. Therefore rvc (G= 3 Subcase 2.2: For = 8k+, k ε N Let G be a graph with vertex set V (G = V U where V = {v i ; i 2k+3} ad ; i 2k+3} ad V 3 ; i 4k+5}. Defie a vertex colorig c 3 : V(G [, ] as follows. c(v i = for ( i 2k+3 ad c(u i = for ( i 4k+5 ad assigig the remaiig colors to the vertices of V (G -V -V 3 as follows, c(u i =i for ( i 2k+3. Without loss of geerality, let u=v ad v=v ad P:u-v is a diameter path. Assume that x, x are iteral vertices i the diameter path have distict colors where x = u, x 2 =u 2.x =u -. Clearly the Path P is a raibow path. Ay path of legth less tha diam (G have distict colors is raibow vertex coected. The rvc (G If we colorig the vertices fewer tha 3 colors, defie a vertex colorig c: V(G [, 3 ] As follows. Assigig color to the vertices of degree exactly 2.c(u i =where i=4k+5,k.c(u i =i iε [,-], i is odd, c(u i = if i= ad c(u 2k =2k, k Let P l be the path of legth exactly diam (G, sice the vertices u ad u have same color ad we ote that these two vertices are i the iteral vertices of the path P l, there exists o raibow path. Let P be the aother path where the vertices u ad u 2 are i the ier vertices have same color, the there exists o raibow path. Hece it is ot possible to take 3 fewer tha. Thus rvc (G 3 If we have vertex colorig more tha colors, the every path is raibow vertex coected with maximum colors chose which cotradicts the defiitio of raibow vertex coectio umber. Thus by choosig rvc (G = 3 colors will be the miimum, which satisfies the defiitio of raibow vertex coectio umber. Thus rvc (G = Figure 4. Cetral graph of comb graph. 8. Coclusio The differece betwee raibow vertex coectio umber for lie graph ad middle graph of Bi-star Graph is zero; whereas raibow vertex coectio umber for cetral graph is 2 ad total graph of Bi-star graph is same if order of a graph is 7 ad ad it is same if order of a graph starts with 5 follows arithmetic progressio with commo differece 4. This results ca be exteded by itroducig radic idex as a method of comparig the radic idex ad raibow vertex coectio umber of lie, middle, cetral ad total graph of Bi-star graph. The applicatios ca be raised with Bi-star Graph ca be treated as chemical graph problem as comparig raibow vertex coectio umber with chemical ad physical properties. 9. Refereces. Chartrad G, Johs GL, Mckeo KA, Zhag P. Raibow coectio i graphs. Math Bohem. 2008; 33(: Chartrad C, Zhaug P. Chromatic Graph Theory. CRC Press; Krivelevich M, Yuster R. The raibow coectio of a graph is (at most reciprocal to its miimum degree. J Graph Theory. 2009; 63(3:85 9. Vol 9 (S December Idia Joural of Sciece ad Techology 5

6 Raibow Vertex Colorig for Lie, Middle, Cetral, Total Graph of Comb Graph 4. Harary F. Graph theory. Naraosa Publishig House; Saha A, Sambroi E, Bogerd J, Schulz RW, Gac FL, Lareyre JJ. The cell cotext iflueces raibow trout goadotropi receptors selectivity. Idia Joural of Sciece ad Techology. 20 Aug; 4(S8: Yao A, Jouao E, Klopp C, Guigue Y. Gee expressio profilig durig goadal differetiatio i raibow trout (Ocorhychusmykiss usig a Next Geeratio Sequecig (NGS approach. Idia Joural of Sciece ad Techology. 20 Aug; 4(S8: Nicol B, Yao A, Jouao E, Brathoe A, Fostier A, Guigue Y. Follistati is expressed alog with aromatase i female goads durig sex differetiatio i the raibow trout. Idia Joural of Sciece ad Techology. 20 Aug; 4(S8:. 8. Valdivia K, Jouao E, Mourot B, Quillet E, Guyomard R, Volff JN, Galiaa-Aroux D, Cauty C, Fostier A, Guigue Y. Masculiizatio i raibow trout carryig the mal mutatio is temperature sesitive. Idia Joural of Sciece ad Techology. 20 Aug; 4(S8:. 9. Kusakabe M, Takei Y, Luckebach JA. Relaxi-3 ad relaxi/isuli-like family peptide receptor 3 i raibow trout: Sites of gee expressio ad chages i messeger RNA levels durig spermatogeesis i testes. Idia Joural of Sciece ad Techology. 20 Aug; 4(S8: 2. 6 Vol 9 (S December Idia Joural of Sciece ad Techology