Prime Cordial Labeling on Graphs

Transcription

1 World Academy of Sciece, Egieerig ad Techology Iteratioal Joural of Mathematical ad Computatioal Scieces Vol:7, No:5, 013 Prime Cordial Labelig o Graphs S. Babitha ad J. Baskar Babujee, Iteratioal Sciece Idex, Mathematical ad Computatioal Scieces Vol:7, No:5, 013 waset.org/publicatio/16747 Abstract A prime cordial labelig of a graph G with vertex set V is a bijectio f from V to {1,,..., V } such that each edge uv is assiged the label 1 if gcd(f(u),f(v)) = 1 ad 0 if gcd(f(u),f(v)) > 1, the the umber of edges labeled with 0 ad the umber of edges labeled with 1 differ by at most 1. I this paper we exhibit some characterizatio results ad ew costructios o prime cordial graphs. Keywords Prime cordial, tree, Euler, bijective, fuctio. I. INTRODUCTION Graph labelig is a strog relatio betwee umbers ad structure of graphs. A useful survey to kow about the umerous graph labelig methods is give by J.A. Gallia [7]. By combiig the relatively prime cocept i umber theory ad cordial labelig [6] cocept i graph labelig, Sudaram, Poraj ad Somasudaram [9] have itroduced the cocept called prime cordial labelig. A prime cordial labelig of a graph G with vertex set V is a bijectio f from V to {1,,..., V } such that each edge uv E { 1 if gcd(f(u),f(v)) = 1 f(uv) = 0 if gcd(f(u),f(v)) > 1 the e(0) e(1) 1 where e(0) is the umber of edges labeled with 0 ad e(1) is the umber of edges labeled with 1. I [4], [5], [9], the followig graphs are proved to have prime cordial labelig: C if ad oly if 6; P if ad oly if 3 or 5; K 1, ( odd); the graph obtaied by subdividig each edge of K 1, if ad oly if 3; bistars; dragos; crows; triagular sakes if ad oly if the sake has at least three triagles; ladders, su graph, kite graph ( >t) ad cocout tree, S (1) : S (), full biary trees from secod level, K ΘC (C ) ad K 1,, for 3. S.K. Vaidya i [11], [1] proved that the square graph of path P is a prime cordial graph for =6ad 8 while the square graph of cycle C is a prime cordial graph for 10. Also they show that the shadow graph of K 1, for 4 ad the shadow graph of B, are prime cordial graphs, certai cycle related graphs, the graphs obtaied by mutual duplicatio of a pair of edges as well as mutual duplicatio of a pair of vertices from each of two copies of cycle C admit prime cordial labelig. I [8] Haque proved that prime cordial labelig of geeralized Peterse graph. Also i [3], prime cordial labelig for some class of cactus graphs ad discuss with duality of prime cordial labelig. Sudaram ad Somasudaram [10] ad Youssef observed that for >3, K is ot prime cordial provided that the iequality ϕ() + ϕ(3) + + ϕ() >(1)/4 + 1 is valid S. Babitha ad J. Baskar Babujee are with the Departmet of Mathematics, Aa Uiversity, Cheai , Idia babi mit@yahoo.co.i for >3. The defiitios ad resuts we used i our paper is give as follows Theorem 1: [13] I ay biary edge labelig the followig are equivalet. { (i) e(0) e(1) 1 q (ii) e(1) = if q is eve q or q 1 if q is odd where x deotes the smallest iteger greater tha or equal to x. (iii) q j=1 e j +(q mod )e d = q where e d is the biary label of a dummy edge which is itroduced oly whe q is odd. Defiitio 1: [] If G 1 (p 1,q 1 ) ad G (p,q ) are two coected graphs, G 1 ôg is obtaied by superimposig ay selected vertex of G o ay selected vertex of G 1. The resultat graph G = G 1 ôg cosists of p 1 + p 1 vertices ad q 1 + q edges. II. CHARACTERIZATION RESULTS ON PRIME CORDIAL GRAPHS I this sectio we give some characterizatio results for prime cordial graphs. Theorem : A (p, q)-graph G is prime cordial. Let f : V (G)to{1,,...,p} be a prime cordial labelig the for each edge uv E(G), f(u)+f(uv)+f(v) p f(v i )deg(v i )+ q = p f(v i )deg(v i )+ ( q or q ) 1 if q is eve if q is odd Proof: Let f be a prime cordial labelig of G(p, q). Cosider f(u)+f(uv)+f(v) = f(u)+ f(uv)+ f(v) Sice f is a bijective fuctio, each vertex label f(u), u V (G) occurs exactly deg(u) times oce i the summatio ad each label uv E(G) admits biary labelig {0, 1} by the cocept of prime cordial which implies f(uv) = e(1). Sice by (ii) of 1.1, = p f(v i )deg(v i )+e(1) Iteratioal Scholarly ad Scietific Research & Iovatio 7(5) scholar.waset.org/ /16747

2 World Academy of Sciece, Egieerig ad Techology Iteratioal Joural of Mathematical ad Computatioal Scieces Vol:7, No:5, 013 Iteratioal Sciece Idex, Mathematical ad Computatioal Scieces Vol:7, No:5, 013 waset.org/publicatio/16747 = f(u)+f(uv)+f(v) p f(v i )deg(v i )+ q p f(v i )deg(v i )+ ( q or q ) 1 if q is eve if q is odd Corollary 1: Let G be a r-regular prime cordial graph o p vertices ad S = f(u)+f(uv)+f(v) the S = { r p(p+1) + q if q is eve r p(p+1) + ( q or q 1) if q is odd Corollary : If a (p, q)-graph G is Prime cordial ad S = f(u)+f(uv)+f(v) the p(p +1) S p f(v i )deg(v i 1) + q = p f(v i )deg(v i 1) + ( q or q ) 1 if q is eve if q is odd Theorem 3: If G is a prime cordial graph, the G e is also prime cordial (i) for all e E(G) whe q is eve (ii) for some e E(G) whe q is odd Proof: Cosider the prime cordial graph G with p vertices ad q edges. Case (i): whe q is eve Let q be the eve size of the prime cordial graph G. The it follows that e(0) = e(1) = q/. Lete be ay edge i G which is labeled either 0 or 1. The i G e, we have either e(0) = e(1)+1 or e(1) = e(0)+1 ad hece e(0) e(1) 1. Thus G e is prime cordial for all e E(G). Case (ii): whe q is odd Let q be the odd size of the prime cordial graph G. The it follows that either e(0) = e(1) + 1 or e(1) = e(0) + 1. If e(0) = e(1) + 1 the remove the edge e which is labeled as 0 ad if e(1) = e(0) + 1 the remove the edge e which is labeled 1 from G. The it follows that e(0) = e(1). Thus G e is prime cordial for some e E(G). Remark 1: G is a prime cordial graph. Similarly we ca prove prime cordiality for the graph which is obtaied by addig oe edge to G. It becomes a iterestig problem to ivestigate the maximum umber of possible edges that ca be costructed i a graph with vertices havig prime cordial labelig property. We iterpret this problem i umber theory to fid total umber of relatively prime pair of itegers i the set {1,, 3,...,} ad fid the maximum umber of edges i prime cordial graph usig Euler s phi fuctio ϕ(). I [1], the total umber of relatively prime pairs for the set S = {1,, 3,...,} is give by S = k= ϕ(k). For vertices, the umber of possible edges labeled with 1 is e(1) = k= ϕ(k) the the umber of possible edges labeled with 0 is e(0) = C k= ϕ(k). By the defiitio of prime cordiality, if e(0) is C k= ϕ(k) the e(1) is at most C k= ϕ(k)+1. Theorem 4: A maximum umber of edges i a simple prime cordial graph with vertices is +1+ k= ϕ(k). Proof: Cosider a graph G(V,E) with vertices is a simple graph with o loops ad parallel edges. The vertices {v 1,v,...,v } are labeled with the itegers {1,,...,} such that each edge v i v j E is assiged the label 1 if gcd(f(u),f(v)) = 1 ad 0 if gcd(f(u),f(v)) > 1, the the umber of edges labeled with 0 ad the umber of edges labeled with 1 differ by at most 1. Number of relatively prime pairs for the set {1,,...,} is k= ϕ(k). For vertices, the umber of possible o relatively prime pairs is C k= ϕ(k). By the defiitio of prime cordial, if e(0) is C k= ϕ(k) the e(1) is at most C k= ϕ(k)+1. For ay prime cordial graph, the maximal umber of edges is e(0) + e(1) = C ϕ(k)+c ϕ(k)+1 k= =C+1 ϕ(k) k= = ( 1) + 1 = +1 ϕ(k) k= ϕ(k). k= k= Hece maximum umber of edges i a simple prime cordial graph with vertices is +1 k= ϕ(k). III. PRIME CORDIAL LABELING FOR G 1 ôg I the followig theorems we cosider the prime cordial graph G ad glue a vertex of some class of graphs to oe of the selected vertex of G ad check whether the ew graph retai the property of prime cordial. For that we eed the theorem give by [13] wieslet. Theorem 5: If G has prime cordial labelig the, GôK 1,m admits prime cordial labelig for (i) m is eve ad G is of ay size q. (ii) m is odd ad (a) G is of eve size q. (b) G is of odd size q with e(1) = q ad order is odd. (c) G is of odd size q with e(1) = q 1 ad order is eve. Proof: Let G(p, q) be a prime cordial graph. Let w V be the vertex whose label is f(w) =. Cosider the star K 1,m with vertex set {x 0,x i : 1 i m} ad edge set {x 0 x i : 1 i m}. We superimpose the vertex x 0 of the star K 1,m graph o the vertex w V of G. Now we defie the ew graph called G 1 = GôK 1,m with vertex set V 1 (G 1 ) = V (G) {x i : 1 i m} ad E 1 (G 1 )=E(G) {vx i :1 i m}. Cosider the bijective fuctio g : V 1 {1,,...,p,p+1,p+,...,p+m} defied Iteratioal Scholarly ad Scietific Research & Iovatio 7(5) scholar.waset.org/ /16747

3 World Academy of Sciece, Egieerig ad Techology Iteratioal Joural of Mathematical ad Computatioal Scieces Vol:7, No:5, 013 Iteratioal Sciece Idex, Mathematical ad Computatioal Scieces Vol:7, No:5, 013 waset.org/publicatio/16747 by g(v i )=f(v i ); 1 i p By our costructio, f(w) =g(x 0 )=. g(x i )=p + i, for1 i m. We have to show that the graph G 1 = GôK 1,m has prime cordial labelig i various cases. Case (i): m is eve ad G is of ay size q The edge set is defied as g(uv) =f(uv) for all uv E(G). If G is of eve order the g(wx i 1 )=1for 1 i m/; g(wx i )=0for 1 i m/. If G is of odd order the g(wx i 1 )=0for 1 i m/; g(wx i )=1for 1 i m/. Hece the total umber of edges labeled with 1 s are give by e(1) = m/ ad the total umber of edges labeled with 0 s are give by e(0) = m/ i the edge set {wx i :1 i m}. Also give that G is already prime cordial. Hece by 1.1, oe ca easily verify that the edge set E 1 (G 1 ) satisfy the coditio e(0) e(1) 1 for the graph G 1 = GôK 1,m. Case (ii): m is odd Subcase (i): G is of eve size q The bijective fuctio g : V 1 {1,,...,p,p +1,p +,...,p+ m} defied as give i above. The edge set is defied as g(uv) =f(uv) for all uv E(G). If G is of eve order the g(wx i 1 )=1for 1 i m ; g(wx i )=0for 1 i m 1. The total umber of edges labeled with 1 s are give by e(1) = m ad the total umber of edges labeled with 0 s are give by e(0) = m 1. SiceG is prime cordial labelig of eve size, e(1) = q ad e(0) = q. Hece for the graph G 1 = GôK 1,m, the total umber of edges labeled with 1 s are give by e(1) = q + m ad the total umber of edges labeled with 0 s are give by e(0) = q + m 1. Therefore the total differece betwee 1 s ad 0 s is give by e(1) e(0) = q + m ( q + m ) =1. If G is of odd order the g(vx i 1 )=0for 1 i m ; g(vx i ) = 1 for 1 i m 1. The total umber of edges labeled with 1 s are give by e(1) = m 1 ad the total umber of edges labeled with 0 s are give by e(0) = m. Sice G is prime cordial labelig of eve size, e(1) = q ad e(0) = q. Hece for the graph G 1 = GôK 1,m, the total umber of edges labeled with 1 s are give by e(1) = q + m 1 ad the total umber of edges labeled with 0 s are give by e(0) = q + m. Therefore the total differece betwee 1 s ad 0 s is give by e(1) e(0) = q + m 1 ( q + m ) =1. Subcase (ii): G is of odd size q with e(1) = q ad order is odd. The bijective fuctio g : V 1 {1,,...,p,p +1,p +,...,p + m} defied as give i above. The edge set is defied as g(uv) =f(uv) for all uv E(G). g(wx i 1 ) = 0 for 1 i m ; g(wx i) = 1 for 1 i 1. The total umber of edges labeled with 1 s are give by e(1) = m 1 ad the total umber of edges labeled with 0 s are give by e(0) = m.siceg is prime cordial labelig of odd size q with e(1) = q,thee(0) = q 1. Hece for the graph G 1 = GôK 1,m, the total umber of edges labeled with 1 s are give by e(1) = q + m 1 ad the total umber of edges labeled with 0 s are give by e(0) = q 1+ m. Therefore the total differece betwee 1 s ad 0 s is give by e(1) e(0) = q + m 1 ( q + m 1) =0. Subcase (iii): G is of odd size q with e(1) = q 1 ad order is eve. The bijective fuctio g : V 1 {1,,...,p,p +1,p +,...,p + m} defied as give i above. The edge set is defied as g(uv) =f(uv) for all uv E(G). g(wx i 1 ) = 1 for 1 i m ; g(wx i) = 0 for 1 i m 1. The total umber of edges labeled with 1 s are give by e(1) = m ad the total umber of edges labeled with 0 s are give by e(0) = m 1. Sice G is prime cordial labelig of odd size q with e(1) = m 1, the e(0) = q. Hece for the graph G 1 = Gˆ0K 1,m, the total umber of edges labeled with 1 s are give by e(1) = q 1+ m ad the total umber of edges labeled with 0 s are give by e(0) = q + m 1. Therefore the total differece betwee 1 s ad 0 s is give by e(1) e(0) = q 1+ m ( q + m 1) =0. Hece usig the above cases Gˆ0K 1,m has prime cordial labelig with the above coditios. Theorem 6: If G has prime cordial labelig the, Gˆ0P m admits prime cordial labeliag for (i) m is eve ad G is of size q. (ii) m is odd ad (a) G is of eve size q. (b) G is of odd size q with e(1) = q ad order is odd. (c) G is of odd size q with e(1) = q 1 ad order is eve. Proof: Let G(p, q) be a prime cordial graph. Let w V be the vertex whose label is f(w) =. Cosider the path P m with vertex set {x i :1 i m} ad edge set {x i x i+1 :1 i m 1}. We superimpose oe of the pedet vertex of the path P m say x 1 o the vertex w V of G. Now we defie the ew graph called G 1 = Gˆ0P m with vertex set V 1 (G 1 ) = V (G) {x i : 1 i m} ad E 1 (G 1 ) = E(G) {wx } {x i x i+1 : i m 1}. Cosider the bijective fuctio g : V 1 {1,,...,p,p +1,p +,...,p + m 1} defied by g(v i )=f(v i ); 1 i p By our costructio, f(w) =g(x 1 )=. Whe p is odd ad m is eve, g(x i ) = p + i 3, for i m+ ; g(x m +i )=p + m (i 1), for i m. Whe p is odd ad m is odd, g(x i ) = p + i 3, for i m ; g(x m +i )=p + m i +3,for i m Whe p is eve ad m is eve, g(x i ) = p + (i 1), for i m ; g(x ( m 1)+i) =p + m i +3,for i m +1. Whe p is eve ad m is odd, g(x i ) = p + (i 1), for i m ; g(x m +i )=p + m (i 1), for i m. Iteratioal Scholarly ad Scietific Research & Iovatio 7(5) scholar.waset.org/ /16747

4 World Academy of Sciece, Egieerig ad Techology Iteratioal Joural of Mathematical ad Computatioal Scieces Vol:7, No:5, 013 Iteratioal Sciece Idex, Mathematical ad Computatioal Scieces Vol:7, No:5, 013 waset.org/publicatio/16747 Usig the above labelig ad similar method of Theorem??, oe ca easily verify that the graph G 1 = GôP m has prime cordial labelig, all the cases give i hypothesis. Theorem 7: If G has prime cordial labelig the, GôF 1,m admits prime cordial labelig for (i) m is eve ad (a) G is of eve size q. (b) G is of odd size with e(1) = q 1. (ii) m is odd ad (c) G is of eve size q ad odd order. (d) G is of odd size q with e(1) = q ad order is odd. Proof: Let G(p, q) be a prime cordial graph. Let w V Fig. 1. GôF 1,5 be the vertex whose label is f(w) =. Cosider the Fa F 1,m with vertex set {x 0,x i : 1 i m} ad edge set {x 0 x i : 1 i m} {x i x i+1 : 1 i m 1}. We superimpose the vertex x 0 of the Fa F 1,m graph o the vertex w V of G. Now we defie the ew graph called G 1 = GôF 1,m with vertex set V 1 (G 1 ) = V (G) {x 0,x i : 1 i m} ad E 1 (G 1 ) = E(G) {wx i : 1 i m} {x i x i+1 : 1 i m 1}. Cosider the bijective fuctio {1,,...,p,p +1,p +,...,p + m} defied g : V 1 by g(v i )=f(v i ); 1 i p By our costructio, f(w) =g(x 0 )=. Whe p is odd ad m is eve, g(x i )=p+i 1,for1 i m ; g(x m +i )=p+m (i 1), for 1 i m. Whe p is odd ad m is odd, g(x i ) = p + i 1, for 1 i m ; g(x m +i )=p + m i +1,for1 i m. Whe p is eve ad m is eve, g(x i )=p +i, for1 i m ; g(x m +i )=p + m i +1, for 1 i m. Whe p is eve ad m is odd, g(x i ) = p +i, for 1 i m ; g(x m +i ) = p + m (i 1), for i m. We have to show that the graph G 1 cordial labelig i various cases. Case (i): m is eve Subcase (i): G is of eve size q. The edge set is defied as = GôF 1,m has prime g(uv) =f(uv) for all uv E(G). For ay p, g(vx i )=0for 1 i m/; g(vx i )=1for m +1 i m ad g(x i x i+1 ) = 0 for 1 i m 1; g(x ix i+1 ) = 1 for m i m 1. Hece the total umber of edges labeled with 1 s are give by e(1) = m ad the total umber of edges labeled with 0 s are give by e(0) = m 1. SiceG is prime cordial labelig of eve size, e(1) = q ad e(0) = q. Hece for the graph G 1 = GôF 1,m, the total umber of edges labeled with 1 s are give by e(1) = q + m ad the total umber of edges ad 0 s is give by e(0) = q + m 1. Therefore the total differece betwee 1 s ad 0 s is give by e(1) e(0) = q + m ( 1 + m 1) =1. Subcase (ii): G is of odd size with e(1) = q 1. The edge set is defied as i subcase (i). Sice G is prime cordial labelig of odd size with e(1) = q 1 the e(0) = q. Hece for the graph G 1 = GôF 1,m, the total umber of edges labeled with 1 s are give by e(1) = q 1+m ad the total umber of edges labeled with 0 s are give by e(0) = q + m 1. Therefore the total differece betwee 1 s ad 0 s is give by e(1) e(0) = q 1+m ( q + m 1) =0. Case (ii): m is odd. Subcase (i): G is of eve size q ad order is odd. The bijective fuctio g : V 1 {1,,...,p,p +1,p +,...,p+ m} defied as give i above. The edge set is defied as g(uv) =f(uv) for all uv E(G). g(vx i ) = 0 for 1 i m ; g(vx i) = 1 for m +1 i m ad g(x i x i+1 )=0for 1 i m 1; g(x ix i+1 )=1for m i m 1. The total umber of edges labeled with 1 s are give by e(1) = m 1 ad the total umber of edges labeled with 0 s are give by e(0) = m. SiceG is prime cordial labelig of eve size, e(1) = q ad e(0) = q. Hece for the graph G 1 = GôF 1,m, the total umber of edges labeled with 1 s are give by e(1) = q + m 1 ad the total umber of edges labeled with 0 s are give by e(0) = q + m. Therefore the total differece betwee 1 s ad 0 s is give by e(1) e(0) = q + m 1 ( q + m)] =1. Subcase (ii): G is of odd size q with e(1) = q is odd. The edge set is defied as same as the subcase (i). ad order Sice G is prime cordial labelig of odd size q with e(1) = q, the e(0) = q 1. Hece for the graph G 1 = GôF 1,m, the total umber of edges labeled with 1 s are give by e(1) = q + m 1 ad the total umber of edges labeled with 0 s are give by e(0) = q 1+m. Therefore the total differece betwee 1 s ad 0 s is give by e(1) e(0) = q + m 1 ( q 1+m) = 0. Hece the graph GôF 1,m has prime cordial labelig with above coditios. Iteratioal Scholarly ad Scietific Research & Iovatio 7(5) scholar.waset.org/ /16747

5 World Academy of Sciece, Egieerig ad Techology Iteratioal Joural of Mathematical ad Computatioal Scieces Vol:7, No:5, 013 Iteratioal Sciece Idex, Mathematical ad Computatioal Scieces Vol:7, No:5, 013 waset.org/publicatio/16747 IV. PRIME CORDIAL LABELING FOR TREE RELATED GRAPHS Theorem 8: The tree K 1,m P (, m > ) obtaied by replacig each edge of K 1,m by equal legth of path has prime cordial labelig. Proof: Cosider the tree K 1,m P with vertex set V = {v i : i = 1,, 3,...,m +1} ad the edge set E = {E 1 E E 3 E 4 E m E m+1 } where E 1 = {v i v i+1 : i }, E = {v i v i+1 : + i },..., E m = {v i v i+1 : (m 1) + i m} ad E m+1 = {v 1 v (i 1)+ :1 i m}. We prove this theorem i three cases. Case (i): For eve m ad ay value of. The bijective fuctio f : V {1,,...,m+1} defied as f(v i )=i for 1 i m+1 ; m+1 +1 ) = m + 1; m+1 + ) = 1; +5 )=5; ) = i 3 for 7 i m+1 +3 )=3; m+1 +4 )=9; m+1 m+1 +6 ) = 7; m+1 m+1 ; +i By the above labelig, the m edges have label 0 ad m edges have label 1. Hece the differece betwee the total umber of edges labeled with 1 s ad 0 s is give by m m =0. Case (ii): For odd m ad eve. The bijective fuctio f : V {1,,...,m+1} defied as f(v i )=i for 1 i m+1 ; m+1 +1 )=1; m+1 + )=3; m+1 m+1 +4 ) = 5; m+1 +5 ) = 7; m+1 i 1 for 6 i m+1 ; +3 )=9; +i ) = By the above labelig, the m edges have label 0 ad m edges have label 1. Hece the differece betwee the total umber of edges labeled with 1 s ad 0 s is give by m m =0. Case (iii): For odd m ad odd. The bijective fuctio f : V {1,,...,m+1} defied as f(v i )=i for 1 i m+1 ; m+1 +i )=i 1 for 6 i ( ) m+1 ; By the above labelig, the m 1 edges have label 0 ad m 1 edges have label 1. Hece the differece betwee the total umber of edges labeled with 1 s ad 0 s is give by ( m ) ( m +1) =1. Hece the tree K 1,m P (, m > ) has prime cordial labelig. Theorem 9: The tree P P m ( 4) obtaied by replacig each vertex of P by equal legth of path has prime cordial labelig. Proof: Cosider the tree P P m with the vertex set V = {v 1,v,...,v m, v m+1,..., v m,v m+1,...,v ( 1)m,v ( 1)m+1,...,v m } ad the edge set E = E 1 E E E +1 where E 1 = {v i v i+1 : 1 i m 1}; E = {v i v i+1 : m + 1 i m 1};...; E 1 = {v i v i+1 :( )m +1 i ( 1)m 1}; E = {v i v i+1 : ( 1)m +1 i m 1} ad E +1 = {v im+1 v (i+1)m+1 :0 i }. We prove this theorem i three cases. Case (i): For ay value of ad m = The bijective fuctio f : V {1,,...,m} defied as f(v i 1 )=i for 1 i ; f(v i )=i 1 for 1 i. I view of the labelig patter defied above, oe ca easily verify that e(0) e(1) =1. Case (ii): For eve ad ay value of m> The bijective fuctio f : V {1,,...,m} defied as f(v i )=i for 1 i m ; f(v m +i )=i 1 for 1 i m. By the defiitio of prime cordial ad the patter of labelig give above, we have the edges i the sets E 1 E E / ad the edges i the set {v im+1 v (i+1)m+1 :0 i 4 } have edge label 0. The edge v ( 1)v ( ) have iduced label 1. Let E +1 = {v im+1 v (i+1)m+1 :0 i }. The vertices from v 1 to v ( 1) i E +1 have eve umbers ad the vertices from v ( ) to v ( ) v ( 1) have odd umbers. Claim: Prove that the labelig f for ay edge uv {v im+1 v (i+1)m+1 : +1 i } is 1 (i.e, f(u) ad f(v) is relatively prime). For that cosider the labelig of {v ( )m+1,v ( )m+m+1,v ( )m+m+1,...,v ( 1)m+1} be {1, m +1, 4m +1,..., ((( ) ) ) 1 m +1} which is icreasig order. Suppose the labelig f for ay edge uv {v im+1 v (i+1)m+1 : +1 i } is ot relatively prime i.e., gcd(f(u),f(v)) > 1. The for ay iteger p, f(u) 0 mod p ad f(v) 0 mod p f(u) f(v) 0 mod p (1) Sice both f(u) ad f(v) are odd, p is also odd ad accordig to our labelig, f(u) f(v) 0 mod () By (1) ad (), p f(u) f(v) ad / f(u) f(v) p is a multiple of p is eve. Which is cotradictio to the fact that p is odd. So the labelig f for ay edge uv {v im+1 v (i+1)m+1 : +1 i } is relatively prime. Therefore the total umber of edges havig edge label 0 is m 1. The remaiig m edges i the edge set E have edge label 1. Hece the differece betwee the total umber of edges labeled with 1 s ad 0 s is give by ( m 1) m =1. Case (iii): For odd ad ay value of m>. Subcase (i): For m is eve The bijective fuctio f : V {1,,...,m} defied as f(v i )=i for 1 i m ; f(v m +i )=m i for 1 i m ; f(v m+i )=i 1 for 1 i m m ; By the above labelig, the total umber of edges havig edge label 0 is e(0) = m 1 ad the total umber of edges havig edge label 1 is e(1) = m. Hece the differece betwee the total umber of edges labeled with 1 s ad 0 s is give by e(1) e(0) = ( m 1) m =1. Subcase (ii): For m is odd. The bijective fuctio f : V {1,,..., V } defied as f(v i )=i for 1 i m ; Iteratioal Scholarly ad Scietific Research & Iovatio 7(5) scholar.waset.org/ /16747

6 World Academy of Sciece, Egieerig ad Techology Iteratioal Joural of Mathematical ad Computatioal Scieces Vol:7, No:5, 013 Iteratioal Sciece Idex, Mathematical ad Computatioal Scieces Vol:7, No:5, 013 waset.org/publicatio/16747 m +i )=m (i 1), for1 i m. m+1 ) = 1; m+ ) = 3; m+4 )=5; m+i )=i 1; for6 i m m ; m+5 )=7; m+3 ) = 9; By the above labelig, the total umber of edges havig edge label 0 is e(0) = m ad the total umber of edges havig edge label 1 is e(1) = m. Hece the differece betwee the total umber of edges labeled with 1 s ad 0 s is give by e(1) e(0) = m m = 0. The edge v ( 1)m+1v ( )m+1 have iduced label 1. Let E +1 = {v im+1 v (i+1)m+1 : 0 i }. The edges from v 1 to v ( 1)m+1 i E +1 have eve umbers ad the edges from v ( )m+1 to v m+( 1)m+1 have odd umbers. We ca prove that the labelig f for ay edge uv {v im+1 v (i+1)m+1 : +1 i + } is relatively prime similar to the claim give i case (ii). Hece P P m has prime cordial labelig. Corollary 3: The comb graph P K has prime cordial labelig. V. CONCLUSION Accordig to literature survey, more work has bee doe i prime cordial labelig for cycle related graphs. I our work we determie the prime cordial labelig for certai classes of trees ad also exhibit some characterizatio results ad ew costructios of prime cordial graph. ACKNOWLEDGMENT The first author Ms. S. Babitha is grateful to the Uiversity Grats Commissio (New Delhi, Idia) for their fiacial support, uder Research Fellowship i Sciece for Meritorious Studets Research Scholars (Sactio No: F.4-1/006(BSR)/11-008(BSR)). REFERENCES [1] J. Baskar Babujee, Euler s Phi Fuctio ad Graph Labelig, It. J. Cotemp. Math. Scieces, 5(0) (010), [] J. Baskar Babujee ad S. Babitha, New Costructios of Edge Bimagic Graphs from Magic Graphs, Applied Mathematics, (011), [3] S. Babitha ad J. Baskar Babujee, Prime Cordial Labelig ad Duality, Israel joural of Mathematics, Commuicated. [4] J. Baskar Babujee ad L. Shobaa, Prime Cordial Labelig, Iteratioal Review of Pure ad Applied Mathematics, 5() (009), [5] J. Baskar Babujee ad L. Shobaa, Prime ad Prime Cordial Labelig, It. J. Cotemp. Math. Scieces, 5(47) (010), [6] I. Cahit, Cordial graphs: A weaker versio of graceful ad harmoious graphs, Ars Combi, 3 (1987), [7] J.A. Gallia, A Dyamic Survey of Graph Labelig, Electroic Joural of Combiatorics, 18 (011), # DS6. [8] Haque, Kh.Md. Momiul, Xiaohui, Li, Yuasheg, Yag, Pigzhog, Zhao, O the Prime cordial labelig of geeralized Peterse graph, Utilitas Mathematica, 8 (010), [9] M. Sudaram, R. Poraj ad S. Somasudram, Prime cordial labelig of graphs, Idia Acad. Math., 7 (005), [10] M. Sudaram, R. Poraj, ad S. Somasudram, Total product cordial labelig of graphs, Bull. Pure Appl. Sci. Sect. E Math. Stat., 5 (006), [11] S.K. Vaidya, N.H. Shah, Some New Families of Prime Cordial Graphs, Joural of Mathematics Research, 3(4) (011), November. [1] S.K. Vaidya, P.L. Vihol, Prime cordial labelig for some cycle related graphs, It. J. Ope Problems Compt. Math., 3(5) (010), Dec. [13] D. Wiselet, T. Nicholas, A Liear Model of E-Cordial, Cordial, H- Cordial ad Geuiely Cordial Labelligs, Iteratioal Joural of Algorithms, Computig ad Mathematics, 3(4) (010), November. S. Babitha received her M.Sc., degree i Mathematics from Aa Uiversity, Cheai, Idia, i 007 ad M.Phil degree from Uiversity of Madras, Cheai, Idia i 008. Curretly she is workig toward her Ph.D. degree i the area of Graph Labelig at Departmet of Mathematics, Aa Uiversity Cheai, Idia. Her Research iterest is graph labelig ad its applicatios. J. Baskar Babujee received M.Sc., M.Phil., from Madras Christia College, Cheai i 1991 ad 199 respectively. He received his Ph.D., degree i 004 from Jawaharlal Nehru Techological Uiversity, Hyderabad, Idia. He has 160 publicatios i academic jourals ad iteratioal coferece proceedigs. He is curretly a Associate Professor i the Departmet of Mathematics, Aa Uiversity, Cheai, Idia. His curret research iterest icludes Graph theory, Cryptography, Formal laguages ad Automata theory. Iteratioal Scholarly ad Scietific Research & Iovatio 7(5) scholar.waset.org/ /16747