K-Total Product Cordial Labelling of Graphs

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1 Avalable at Appl. Appl. Math. ISSN: Vol. 7, Issue (December ), pp 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

2 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.

3 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.

4 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).

5 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.

6 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.

7 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:.

8 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, 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, 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,

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