4-PRIME CORDIAL LABELING OF SOME DEGREE SPLITTING GRAPHS

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1 Iteratioal Joural of Maagemet, IT & Egieerig Vol. 8 Issue 7, July 018, ISSN: Impact Factor: Joural Homepage: Double-Blid Peer Reviewed Refereed Ope Access Iteratioal Joural - Icluded i the Iteratioal Serial Directories Idexed & Listed at: Ulrich's Periodicals Directory, U.S.A., Ope J-Gage as well as i Cabell s Directories of Publishig Opportuities, U.S.A 4-PRIME CORDIAL LABELING OF SOME DEGREE SPLITTING GRAPHS Rajpal Sigh R. Poraj R.Kala Keywords: Complete graph; cycle; coroa; bistar; jelly fish Abstract Let G be a (p,q) graph. Let f : V(G) {1,,,k} be a map. For each edge uv, assig the label gcd (f(u),f(v)). f is called k-prime cordial labellig of G if v f (i)-v f (j) 1, i,j {1,,,k} ad ef(0) ef(1) 1 where v f (x) deotes the umber of vertices labelled with x,e f (1) ad e f (0) respectively deote the umber of edges labelled with 1 ad ot labelled with 1. A graph with a k-prime cordial labellig is called a k-prime cordial graph. I this paper we ivestigate 4-prime cordial labellig behaviour of degree splittig graph of path, jelly fish, crow ad bistar ad some more graphs. Research Scholar, Departmet of Mathematics, Maomaium Sidaraar Uiversity, Abishekapatti, Tiruelveli-6701, Tamiladu, Idia. Assistat Professor,Departmet of Mathematics,Sri Paramakalyai College, Alwarkurichi ,Tamiladu, Idia Professor,Departmet of Mathematics, Maomaium Sidaraar Uiversity, Abishekapatti, Tiruelveli-6701, Tamiladu, Idia 7

2 1. Itroductio I this paper graphs are fiite, simple ad udirected. Let G be a (p, q) graph where p refers the umber of vertices of G ad q refers the umber of edge of G. The umber of vertices of a graph G is called order of G, ad the umber of edges is called size of G. The cocept of degree splittig graph was itroduced by R. Poraj ad S.Somasudaram i [5]. Let G = (V,E) be a graph with V = S 1 S S t T where each S i is a set of vertices havig at least two vertices ad havig the same degree ad T = V t i 1 S i. The degree Splittig graph of G deoted by DS (G) is obtaied from G by addig vertices w 1,w...,w t ad joiig w i to each vertex of S i (1 i t). Let G 1, G respectively be (p 1, q 1 ), (p, q ) graphs. The coroa of G 1 with G, G 1 G is the graph obtaied by takig oe copy of G 1 ad p 1 copies of G ad joiig the i th vertex of G 1 with a edge to every vertex i the i th copy of G. The bistar B m, is the graph obtaied by makig adjacet the two cetral vertices of K 1,m ad K 1,. Jelly fish graphs J (m, ) obtaied from a cycle C 4 : uvxyu by joiig x ad y with a edge ad appedig m pedet edges to u ad pedet edges to v. I 1987, Cahit itroduced the cocept of cordial labellig of graphs [1]. Sudaram, Poraj, Somasudaram [6] have itroduced the otio of prime cordial labelig. Also they discussed the prime cordial labelig behaviour of various graphs. Recetly Poraj et al. [8], itroduced k-prime cordial labelig of graphs. I [9, 10] Poraj et al. studied the 4-prime cordial labelig behaviour of complete graph, book, flower, mc, wheel, gear, double coe, helm, closed helm, butterfly graph, ad friedship graph ad some more graphs. I this paper we study about the 4-prime cordiality of degree splittig graph of path, jelly fish graph, crow, bistar, subdivisio of a star, subdivisio of bistar ad subdivisio of crow. A biary vertex labelig f : V (G) {0, 1} of graph G with iduced edge labelig f : E(G) {0, 1} defied by f (uv) = f(u)f(v) is called a product cordial labelig if v f (0) v f (1) 1 ad e f (0) e f (1) 1, where v f (0), v f (1) deote the umber of vertices of G havig labels 0, 1 respectively uder f ad e f (0), e f (1) deote the umber of edges of G havig labels 0, 1 respectively uder f. A graph G is product cordial if it admits product cordial labelig [7]. Let x be ay real umber. The x stads for the largest iteger less tha or equal to x ad x stads for smallest iteger greater tha or equal to x. Terms that are ot defied here, follow from Harary [3] ad Gallia []. 8

3 . Mai results Observatio.1. A -prime cordial labelig is a product cordial labelig. [7] Proof: Obviously, sice -Prime cordial labelig produces same vf(x) ad ef(x) { x= 0,1} as i product cordial labelig. Theorem.1. DS(P ) is 4-prime cordial for all. Proof. Let P be the path u 1 u... u. Let V (DS(P )) = V (P ) {u, v} ad E(DS(P )) = E(P ) {uu 1, uv, vu i : i 1}. Clearly DS(P ) has + vertices ad 1 edges. The proof is divided ito four cases. Case 1. 0 (mod 4). Let = 4t. Assig the label to the vertices u 1, u,...u t ad 4 to the verticesu t+1, u t+,..., u t. The assig the label 1 ad 3 alteratively to the remaiig vertices. Fially assig the label, 4 respectively to the vertices u, v. Case. 1 (mod 4). As i case 1 assig the label to the vertices u, v, u i (1 i 1). Fially assig the label 3 to the last vertex u. Case 3. (mod 4). Assig the label to the vertices u, v, u i, (1 i 1) as i case. The assig the label 1 to the vertex u. Fially iterchage the labels of u + ad u +3 Case 4. 3 (mod 4). Assig the label to the vertices u, v, u i, (1 i 1) as i case 3. The assig the label to the vertex u. Theorem.. DS(C K 1 ) is 4-prime cordial for all values of. Proof. Let V (C K 1 ) = {u i, v i : 1 i } ad E(C K1) = {u i u i+1, u i v i :1 i 1} {u u 1, u v }. The graph DS(C K 1 ) is obtaied by addig the ew vertices u, v ad joiig u to u i (1 i ), v to v i (1 i ). We owgive the label to the vertices of DS(C K 1 ). Assig the label, 1 respectively tothe vertices u ad v. Next assig the label to the vertices u 1, u,..., u ad 4 to the vertices u u,...,u 1, v. Next assig the label 1 to the vertices v 1,v,, ad 3 to the 9

4 vertices v v,..., v. The table 1 give below establish the labellig f is a 4-prime cordial labellig. 1, Nature of v f (1) v f () v f (3) v f (4) e f (0) e f (1) is odd is eve Table 1. Theorem.3. If 1, 3 (mod 4), the DS(B, ) is 4-prime cordial. Proof. Let V (DS(B, )) = {u, v, x, y, u i, v i : 1 i } ad E(DS(B, )) ={uv, uy, vy, uu i, vv i, xu i, xv i : 1 i }. Clearly DS(B, ) has + 4 verticesad edges. Case 1. 1 (mod 4). Let = 4t + 1. Assig the label to the vertices u 1, u,..., u t+. Next assigthe label 4 to the vertices u t+3, u t+4,..., u 4t+1, x, y ad u. Next assig thelabel 1 to the vertices v 1, v,..., v t. Fially assig the label 3 to the verticesv t+1, v t+,..., v 4t+1. Case. 3 (mod 4). As i case 1 assig the label to the vertices u, v, x, y, u i, v i (1 i ). Fiallyassig the labels, 4, 1, 3 respectively to the vertices u 1, u, v 1,v. Obviously this vertex labellig is a 4-prime cordial labellig of DS(B, ), 1, 3 (mod 4) follows from Table. Nature of v f (1) v f () v f (3) v f (4) e f (0) e f (1) = 4t+1 t+1 t+ t+1 t+ 8t+1 8t+ = 4t+3 t+ t+3 t+ t+3 8t+3 8t+4 Table. Theorem.4. Degree splittig graph of a subdivisio of a star K 1,, DS(S(K 1, )) is 4-prime cordial for all values of. Proof. Let V (DS(S(K 1, ))) = {u,w i, v i, v,w : 1 i } ad E(DS(S(K 1, ))) ={uw i,w i v i, vv i,ww i : 1 i }. Obviously DS(K 1, ) has + 3 vertices ad 4edges. 30

5 Case 1. 0 (mod 4). Let = 4t, t N. Cosider the vertex u. Assig the label to the vertex u. Next assig the labels 4, 1 to the vertices w, v respectively. We ow cosider the verticesof degree 3. Assig the label to the vertices w 1, w,..., w t ad 4 to the verticesw t+1, w t+,..., w 4t. Now we move to the vertices of degree. Assig the label 1to the vertices v 1, v,..., v t ad 3 to the remaiig vertices v t+1, v t+,..., v 4t. Case. 1 (mod 4). Let = 4t + 1. As i case 1, assig the labels to the vertices u, v, w, v i, w i (1 i 1). Fially assig the labels 4, 3 to the vertices w, v respectively. Case 3. (mod 4). I this case assig the label to the vertices u, v, w, v i, w i (1 i 1) as i case. Next assig the labels, 1 to the vertices w, v respectively. Case 4. 3 (mod 4). As i case 3, assig the labels to the vertices u, v, w, v i, w i (1 i 1). Fially assig the labels 3, 4 to the remaiig vertices v, w respectively. The followig table 3 establish that this vertex labelig f is a 4-prime cordial labelig. Nature of v f (1) v f () v f (3) v f (4) e f (0) e f (1) 4t t+1 t+1 t t+1 8t 8t 4t+1 t+1 t+1 t+1 t+ 8t+1 8t+1 4t+ t+ t+ t+1 t+ 8t+ 8t+ 4t+3 t+ t+ t+ t+3 8t+3 8t+3 Table 3 Theorem.5. DS(J(, )) is 4-prime cordial. 31

6 Proof. Let V (DS(J(, ))) = {u, v, x, y,w 1,w,w 3 } {u i, v i : 1 i }, ad E(DS(J(, )) = {uy, vy, uw 1, uw 3, vw 1, vw 3,w 1 w,w w 3 } {uu i, vv i, xu i, xv i :1 i }. Clearly DS(J(, )) has + 7 vertices ad edges. Case 1. 0 (mod 4). Let = 4t. Assig the label to the vertices u 1, u,..., u t+. Next assig thelabel 4 to the vertices u t+3, u t+4,..., u 4 t. The assig the label 4 to the verticesu, x, y, w 3. Assig the label 1 to the vertex v. Next assig the label 1 to thevertices v 1, v,..., v t. Fially assig the remaiig o-labeled vertices with 3. Case. 1 (mod 4). Let = 4t + 1. I this case, assig the label to the vertices u 1, u,..., u t+3. Next assig the label 4 to the vertices u t+4, u t+5,..., u 4t+1. Next assig the label 4 to the vertices x, u, v, y. Next assig the label 1 to the vertices v 1, v,..., v t+. Fially assig the remaiig o-labeled vertices with 3. Case 3. (mod 4). As i case assig the label to the vertices u i, v i (1 i 1), u, v, x, y, w 1, w, w 3. Fially assig the label 4, 3 respectively to the vertices u, v. Case 4. 3 (mod 4). Assig the label to the vertices u i, v i (1 i 1), u, v, x, y, w 1, w, w 3 as i case 3. Fially assig the label, 1 respectively to the vertices u, v. The followig table 4 establish that the above vertex labelig f is a 4-prime cordial labelig. Nature of v f (1) v f () v f (3) v f (4) e f (0) e f (1) 0 (mod 4) t+1 t+3 t+ t+ 8t+4 8t+4 1 (mod 4) t+ t+3 t+ t+ 8t+6 8t+6 (mod 4) t+ t+3 t+3 t+3 8t+8 8t+8 3 (mod 4) t+3 t+4 t+3 t+3 8t+10 8t+10 3

7 Table 4 Example.1. A 4-prime cordial labelig of J(5, 5) is give i Figure 1. Figure 1 Example.. A 4-prime cordial labelig of J(6, 6) is give i Figure. 33

8 Figure Example.3. A 4-prime cordial labelig of J(7, 7) is give i Figure 3. Figure 3 34

9 Theorem.6. DS(S(B, )) is 4-prime cordial for all values of. Proof. Let V (DS(S(B, ))) = {u, v,w, x,w 1,w } {u i, v i, x i, y i : 1 i } ad E(DS(S(B, ))) = {uu i, u i x i, vv i, v i y i,w 1 x i,w 1 y i,w u i,w v i : 1 i }. It iseasy to verify that DS(S(B, )) has vertices ad edges. Assig thelabel to the vertices u, u i (1 i ). Next cosider the vertices x i, assig the label 4 to the vertices xi (1 i ), w 1, w. Assig the label 3 to the verticesv 1, v, y 1, y,..., y 1. Fially assig the label 1 to all the o-labeled vertices. This vertex labelig f is a 4-prime cordial labelig sice v f (1) = v f (4) = +,v f () = v f (3) = + 1 ad e f (0) = e f (1) = 4 +. Theorem.7. DS(S(C K 1 )) is 4-prime cordial for all values of. Proof. We take the vertex set ad edge set of C K 1 as i Theorem.. Let V (DS(S(C K 1 ))) = V (C K 1 ) {w 1,w,w} {x i, y i : 1 i } ad E(DS(S(C K 1 ))) = {wx i,wy i, y i v i u i x i, u i y i,w 1 v i,w u i : 1 i }. Clearly DS(S(C K 1 )) has 4+3 vertices ad 8 edges. We ow give the 4-prime cordial labelig to this graph. Assig the label to the vertices u i (1 i ) ad 4 to the vertices x i (1 i ). Assig the label 3 to the vertices v i (1 i ) ad to the vertex w 1. Fially assig the label 1 to the o-labeled vertices. If f is this vertex labelig the v f (1) = v f () = v f (3) = + 1, v f (4) = ad e f (0) = e f (1) = 4.This implies f is a 4-prime cordial labelig. Refereces [1] I.Cahit, Cordial Graphs: A weaker versio of Graceful ad Harmoious graphs, Ars combi.,3 (1987) [] J.A.Gallia, A Dyamic survey of graph labelig, The Electroic Joural of Combiatorics,17 (015) #Ds6. [3] F.Harary, Graph theory, Addisio Wesley, New Delhi (1969). [4] M.A.Seoud ad M.A.Salim, Two upper bouds of prime cordial graphs, JCMCC, 75(010) [5] R. Poraj ad S. Somasudaram, O the Degree Splittig graph of a graph, Natioal Academy Sciece Letter, 7(004)

10 [6] M.Sudaram, R.Poraj ad S.Somasudaram, Prime cordial labelig of graphs, J. IdiaAcad. Math., 7(005) [7] M. Sudaram, R. Poraj ad S. Somasudaram, Product cordial labelig of graphs, Bull. Puread Appl. Sci. (Math. & Stat.), 3E (004) [8] R.Poraj Rajpal Sigh, R.Kala ad S. Sathish Narayaa, k-prime cordial graphs, J. Appl.Math. & Iformatics, 34 (3-4) (016) [9] R.Poraj, Rajpal Sigh ad S. Sathish Narayaa, 4-prime cordiality of some classes of graphs, Iteratioal Joural of Mathematical Combiatorics, (017), [10] R.Poraj, Rajpal Sigh ad S. Sathish Narayaa, 4-prime cordiality of some cycle related graphs, Applicatios ad Applied Mathematics, 1(1) (017),