K-Total Product Cordial Labelling of Graphs

Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 93-9466 Vol. 7, Issue (December ), pp. 78-76 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) K-Total Product Cordal Labellng o Graphs R. Ponraj and M. Sundaram (retred) Department o Mathematcs Sr Paramakalyan College Alwarkurch-674, Inda ponrajmaths@ndatmes.com ; sundaram_spkc@redmal.com M. Svakumar Department o Mathematcs Unnamala Insttute o Technology Kovlpatt-685, Inda svamaths.van_r@yahoo.com Receved: December, ; Accepted: August, Abstract In ths paper we ntroduce the k-total Product cordal labellng o graphs. Also we nvestgate the 3-Total Product cordal labellng behavour o some standard graphs. Keywords: Path, Cycle, Star, Comb MSC No.: 5C78. Introducton The graphs consdered here are nte, undrected and smple. The vertex set and edge set o a graph G are denoted by V (G) and E (G) respectvely. The graph obtaned by subdvdng each edge o a graph G by a new vertex s denoted by S (G). The corona G ΘG o two graphs G and G s dened as the graph G obtaned by takng one copy o G (whch has p vertces) and p copes o G and then jonng the th vertex o G to every vertex n the th copy G.Terms not dened here are used n the sense o Harary (969). Rosa (967) ntroduced the concept o β-valued graph and Caht (987) was nstrumental or the ntroducton o a weaker verson o the above concept, known as cordal labellng. Several 78

AAM: Intern. J., Vol. 7, Issue (December ) 79 authors studed cordal graphs [Gallan ()]. Motvated by these dentons, Sundaram et al. (4) ntroduced Product cordal labellng o graphs. Some authors are now workng on Product cordal graphs [Saleh (); Selvaraju (9); Seoud (); Vadya, (), ()] and several varatons o t [Babujee (), Sundaram (5)]. The authors have ntroduced a generalzed orm o Product cordal labellng, known as the k- Product cordal labellng [Ponraj ()]. In ths paper we ntroduce a new concept known as the k-total Product cordal labellng and nvestgate 3-Total Product cordal labellng behavour o some standard graphs.. K-Total Product Cordal Labellng Denton.. Let be a map rom V(G) to {,,..., k-}, where k s an nteger, k V(G). For each edge uv, assgn the label (u) (v) (mod k). s called a K-Total Product cordal labellng o G () (j),, j {,,, k-}, where (x) denotes the total number o vertces and edges labelled wth x (x =,,,, k-). Theorem.. Let G be a (p,q) k-product cordal graph. I p (mod k) or q (mod k) then G s k-total Product cordal. Proo: Case (): p (mod k). Let p= kt. Let be a k- Product cordal labellng o G. Snce s a k-product cordal labellng, v ()=t and e ( ) e ( j), I k-, j k-, where v (x) and e (x) denote the number o vertces and edges respectvely labeled wth x (x =,,,3,, k-). Now ( j) = v ( ) e ( ) ( v ( j) e ( j)) = v ( ) v ( j) e ( ) e ( j) = e ( ) e ( j). Case (): Smlar to () snce e () = e (j). Theorem.3. Any path P n s 3-Total Product cordal.

7 R. Ponraj et al. Proo: Let P n be the Path u u...u n. Case (): n (mod 3). Let n = 3t. Dene (u ) =, t and (u t+ ) =, t. Here, () = t, () = t-, () = t. Thereore, s a 3-Total Product cordal lablng. Case (): n (mod 3). Let n=3t+. Dene (u ) =, t and (u t+ ) =, t+. Snce () = t, ()=t, () = t+, Here, s a 3-Total Product cordal labellng. Case (): n (mod 3). Let n = 3t+. Dene a map as ollows: (u ) =, (u ) =, (u + ) =, t-, (u t++ ) =, t+. Here, () = () = () = t+. Thereore, s a 3-Total Product cordal labellng. Illustraton.4. Fgure. A 3-Total product cordal labellng o P 8. Theorem.5. The Cycle C n s 3-Total product cordal labellng n 3, 6. Proo: Let C n be the cycle u u... u n u. Case (): n (mod 3).n > 6. Let n = 3t, t >. Dene (u ) =, (u ) =, (u + ) =, t-, (u t+ ) =, t. Clearly, ( )= ( ) = () = t. Thereore, s a 3-Total Product cordal labellng.

AAM: Intern. J., Vol. 7, Issue (December ) 7 Case (): n (mod 3). Let n = 3t+. Dene (u ) =, t and (u t+ ) =, t+. Here, () = t+, () = t+. Thereore, s a 3-Total Product cordal labellng. Case (): n (mod 3). Let n = 3t+. Dene (u ) =, t and (u t+ ) =, t+. Here, () = t+, () = t+, () = t+. Thereore, s a 3-Total Product cordal labellng. Case (v): n = 3. Suppose s a Total Product cordal labellng o C 3. Here, sum o the order and sze o C 3 s 6. Clearly, () 3, a contradcton. Case (v): n = 6. Here, sum o the order and sze o C 6 s. I s labelled wth vertex, then () = 3. I labelled wth any two vertces then () 5, whch should not happen. Illustraton.6. Fgure. A 3-Total product cordal labellng o C. Result.7. Ponraj (). Any Star s k-product cordal. Theorem.8. The Star K,n s 3-Total Product cordal n =, (mod 3).

7 R. Ponraj et al. Proo: V (K,n ) = { u,v, n} and E (K,n ) = {uv, n}. Case (): n, (mod 3). The result ollows rom theorem. and.7. Case (): n (mod 3). Let n = 3t+. Here, the sum o order and sze o the star s 6t+3. Clearly, (u). Subcase (): (u) =. Suppose x pendant vertces are labelled wth and y pendant vertces are labelled wth.then n- x-y pendant vertces are labelled wth. Thereore, () = x, () = y+, () = (n-x-y). But, () = () = () = t+. Thereore, x = t+, an mpossblty. Subcase (): (u) =. Smlar to Subcase (), we get a contradcton. Hence K,n s s 3-Total Product cordal labellng n =, (mod 3). Illustraton.9. Fgure 3. A 3-Total product cordal labellng o K,. Remark.. Any star s k-product cordal [Ponraj ()] and hence a k-product cordal graph need not be a k- Total Product cordal graph.

AAM: Intern. J., Vol. 7, Issue (December ) 73 Theorem.. The Comb s 3-Total Product cordal. Proo: Let P n be the path u u u 3... u n. Also, let v be the pendant vertex adjacent to u ( n ) Case (): n (mod 3). Let n=3t. Dene (u ) = (v ) =, t, (u t+ ) =, t, (v t+ ) =, t. Here, () = 4t, ()=4t -, () =4t. Thereore, s a Total Product cordal labellng. Case (): n (mod 3). Let n = 3t+. Dene (u ) =, t, (v ) =, t-, (v t ) =, (v t+ ) =, (u t+ ) =, t+, (v t++ ) =, t. Here, () = () = () =4t +. Thereore, s a Total Product cordal labellng. Case (): n (mod 3). Let n=3t+. Dene (u ) =, t, (v ) =, t+, (u t+ ) =, t+, (v t-+ ) =, t+. Here, () = 4t+, ()=4t+, () =4t+. Thereore, s a Total Product cordal labellng. Theorem.. P n ΘK s 3- Total Product cordal. Proo: Let P n be the path u,u...u n. Let v and w be the pendant vertces whch adjacent to u, n. Dene (u ) =, n, (v ) =, n, (w ) =, n, () = n, ()=n-, () = n. Thereore, s a 3-Total Product cordal.

74 R. Ponraj et al. Illustraton.3. Theorem.4. Fgure 4. A 3-Total product cordal labellng o P 5 ΘK. K +mk s 3-Total Product cordal m = (mod 3). Proo: Let V(K +mk ) = Case (): m (mod 3)., v, u : n and E(K +mk ) = uv, uu, vu : n u Let m=3t. I possble, let there be a 3-Total Product cordal labellng. The sum o the order and sze o K +mk s 9t+3. Thereore, () = () = () =3t+. Clearly, (u) and (v) are not equal to zero otherwse () 3t+. Let x, y be the number o vertces n mk labelled wth and, respectvely. Then 3x = 3t+, a contradcton. Case(): m (mod 3). Let m=3t+. Here, the sum o sze and order s 9t+6. Here, () = () = () =3t+. Let x be the number o vertces n mk labelled wth. Then, 3x=3t+, a contradcton. Case(): m (mod 3). Let m=3t+. Dene (u) = (v ) =, (u ) =, t+, (u t++ ) =, t, (u t++ ) =, t+. Here, () = ()= () = 3t+3. Thereore s 3- Total Product cordal. Theorem.5. S(K,n ) s 3-Total Product cordal. Proo:.

AAM: Intern. J., Vol. 7, Issue (December ) 75 Let V(S(K,n ) )= u Case (): n (mod 3)., u, v : n and E(S(K,n ) ) = uu u v : n,. Let n=3t. Dene (u) =, (u ) =, t, (u t+ ) =, t, (v ) =, t, (v t+ ) =, t, ()= 4t+, () = ()=4t. Hence s 3-Total Product cordal labellng. Case (): n (mod 3). Let n=3t+. Dene (u) =, (u ) =, t, (u t+ ) =, t+, (v ) =, t, (v t+ ) =, t+, ()= 4t+, () = ()=4t+. Hence, s 3-Total Product cordal labellng. Case (): n (mod 3). Let n=3t+. Dene (u) =, (u ) =, t, (u t+ ) =, (u t++ ) =, t+, (v ) =, t, (v t+ ) =, t, (v t+ ) =, (v t++ ) =, t, (v 3t+ ) =, () = ()= ()=4t+3. Hence, s 3-Total Product cordal labellng. 3. Conclusons In ths paper we have explored the cases when a k-product cordal graph become K-Total Product cordal and also studed the k-total Product cordal behavour o some graphs or the specc value k = 3.It shall be nterestng to study the K-Total Product behavour o standard graphs or general k. Reerences Baskar Babujee, J., and Shobana, L. (). Prme and Prme Cordal Labellng or Some Specal Graphs, Int. J. Contemp., Math. Scences, 5, 347-356. Caht, I. (987). Cordal graphs, A weaker verson o graceul and harmonous graphs, Ars Combnatora, 3, -7. Ebrahm, Saleh (). PC-Labellng o a Graph and ts PC- Set, Bulletn o the Insttute o Combnatorcs and ts Applcatons, 58, -. Gallan, J.A. (9). A dynamc survey o graph labellng, The Electroncs J. o Combnatorcs, 6, # DS6. Harary, F. (969). Graph Theory, Addson Wsely, New Delh. Momnul, Haque, Kh. Md., Ln Xaohu, Yang, Yuansheng and Zhao, Pngzhong (). On the Prme cordal labellng o generalzed, Utltas Mathematca, 8,7-79. Ponraj, R., Svakumar, M., and Sundram, M. ( ). k-product cordal labellng o graphs, Int. J. Contemp. Math. Scences, Int. J. Contemp. Math. Scences, Vol. 7,, no. 5, 733-74. Rosa, A. (967). In certan valuaton o vertces o a graph, Theory o graphs (Internatonal Symposum, Rome July 966) Gorden and Breach N.Y and Dunod, Pars, 349-355.

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