Z 4p - Magic labeling for some special graphs

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1 Internatonal Journal of Mathematcs and Soft Computng Vol., No. (0, ISSN Prnt : 49-8 Z 4p - Magc labelng for some specal graphs ISSN Onlne: 9-55 V.L. Stella Arputha Mary Department of Mathematcs, St.Mary s College, Tutcorn E-mal: prsstell@yahoo.com S. Navaneethakrshnan, A. Nagarajan Department of Mathematcs, V.O.C College, Tutcorn E-mal: snk.voc@gmal.com, nagarajan.voc@gmal.com Abstract For any non-trval abelan group A under addton a graph G s sad to be A magc f there exsts a labelng f of the edges of G wth non zero elements of A such that, the vertex labelng f + defned as f + (v = Σf(uv taken over all edges uv ncdent at v s a constant [5]. A graph s sad to be A-magc f t admts an A-magc labelng. In ths paper we prove that splttng graph of a path, trangular snake and book graphs are Z 4 -magc graphs. Also we generalze that they are all Z 4p -magc graphs for any postve nteger p. Keywords: A - magc labelng, Z 4 - magc labelng, Z 4p -magc labelng, Z 4p -magc graphs. AMS Subject Classfcaton(00: 05C78. Introducton In ths paper by a graph G(V, E we mean G s a fnte, smple, undrected graph. The concept of magc labelngs were ntroduced by Sedlacek n 96. Kong, Lee and Sun [4] used the term magc labelng for the labelng of edges wth non negatve ntegers such that for each vertex v, the sum of the labels of all edges ncdent at v s same for all v. For any non-trval abelan group A under addton a graph G s sad to be A magc f there exsts a labelng f of the edges of G wth non zero elements of A such that, the vertex labelng f + defned as f + (v = Σf(uv taken over all edges uv ncdent at v s a constant. In ths paper, we choose Z 4 whch s addtve modulo 4 as the abelan group and we prove the splttng graph of a path, trangular snake, book graph and F n (t are Z 4 -magc graphs. We also prove that they are all Z 4p -magc graphs. Defntons Defnton.. [6] For each pont v of a graph G take a new vertex v and jon v to those ponts of G adjacent to v. The graph thus obtaned s called the splttng graph of G and s denoted as S (G. Defnton.. [] The block - cutpont graph of a graph G s a bpartte graph n whch one partte set conssts of the cut vertces of G and the other has a vertex b for each block B of G. Defnton.. [] A block of a graph s a maxmal connected subgraph that has no cut-vertex. Defnton.4. [] A trangular cactus s a connected graph all of whose blocks are trangles. Defnton.5. [] A trangular snake s a trangular cactus whose block-cutpont graph s a path. 6

2 6 V.L. Stella Arputha Mary, S. Navaneethakrshnan and A. Nagarajan Defnton.6. [] A book wth n pages s defned as the Cartesan product of the complete bpartte graph K,n and a path of length and s denoted by B n. Defnton.7. [] The graph P n + K n s called a fan and t s denoted by F n. Defnton.8. F (t n s the one-pont unon of t fans of length n. Man Results Theorem.. S (P n s Z 4 -magc for n. Proof: Let the vertex set V (S (P n = {v / n} {v / n} and the edge set E(S (P n = {v v + / n } {v v + / n } {v v +/ n }, where v, v,...v n are the new vertces joned correspondng to v, v,...v n of the path P n. Defne f : E(S (P n Z 4 {0} as { } for =, n f(v v + = n f(v v = = f(v n v n f(v v + =, n andf(v v + =, n Then the mappng f + : V (S (P n Z 4 s gven by f + (v = f(v v + f(v v + + f(v v + f(v v +, n f + (v = f(v v + f(v v f + (v n = f(v n v n + f(v n v n f + (v = f(v v + f(v v +, n f + (v = f(v v f + (v n = f(v n v n Clearly, f + (v = f + (v =, n f + (v =, n Thus S (P n admts Z 4 - magc labelng. Hence, S (P n s a Z 4 -magc graph. Example.. Z 4 -magc labelngs of S (P 7 and S (P 6 are gven below. v v v v 4 v 5 v 6 v 7 v v v v 4 v 5 v 6 v 7 Fgure : Z 4 - magc labelng of S (P 7.

3 Z 4p - Magc labelng for some specal graphs 6 v v v v 4 v 5 v 6 v v v v 4 v 5 v 6 Fgure : Z 4 - magc labelng of S (P 6. Theorem.. Trangular snake T n s Z 4 -magc, for n. Proof: Let V (T n = {v / n + } {v / n} and E(T n = {v v + / n} {v v + / n} {v v / n}. V (T n = n + and E(T n = n. Case : n s odd. Defne f : E(T n Z 4 {0} as f(v v + =, (n / f(v v =, < < (n + / f(v j v j = f(v jv j+ =, j n Then f + : V (T n Z 4 s defned as f + (v j = f(v j v j + f(v j v j+ + f(v j v j + f(v j v j, j n f + (v = f(v v + f(v v f + (v n+ = f(v n v n+ + f(v n+ v n f + (v j = f(v j v j + f(v jv j+, j n Then we have, f + (v =, n + f + (v j =, j n. Hence, f + s a constant and t s equal to for all v V (T n. Case : n s even. Defne f : E(T n Z 4 0 by f(v v =, n/ f(v v + =, n/ f(v j v j =, j n f(v jv j+ =, j n. Then f + : V (T n Z 4 s gven by f + (v j = f(v j v j + f(v j v j+ + f(v j v j + f(v j v j ; j n f + (v = f(v v + f(v v ; f + (v n+ = f(v n v n+ + f(v nv n+ ;

4 64 V.L. Stella Arputha Mary, S. Navaneethakrshnan and A. Nagarajan f + (v j = f(v jv j+ + f(v j v j j n. Then we have, f + (v = 0, n + and f + (v j = 0, j n. In both the cases T n admts Z 4 - magc labelng. Hence, T n s Z 4 - magc graph. Example.4. Z 4 - magc labelngs of T n for n = 4 and n = 5 are gven below. v v v v 4 v 5 v v v v 4 v 5 v 6 Fgure : Z 4 - magc labelng of T 5. v v v v 4 v 5 v 6 v v v v 4 v 5 v 6 v 7 Fgure 4: Z 4 - magc labelng of T 6. Theorem.5. The graph B n s Z 4 - magc for all n N. Proof: Let V (B n = {u, v} {u, v / n} and E(B n = {uv} {uu / n} {vv / n} {u v / n}. Case : n s odd. Defne f : E(B n Z 4 {0} by f(uv = f(u v =, n f(uu =, n f(vv =, n. Then f + : V (B n Z 4 s defned by f + (u = f(uv + Σ n =f(uu f + (v = f(uv + Σ n =f(vv f + (u = f(uu + f(u v n f + (v = f(vv + f(u v n.

5 Z 4p - Magc labelng for some specal graphs 65 We have, f + (u =, f + (v =, f + (u =, nandf + (v =. n. Case : n s even. Defne f : E(B n Z 4 {0} by f(uv = f(u v =, n f(uu =, n f(vv =, n Then Clearly, f + (u = = f + (v f + (u = = f + (v n In both the cases B n admts Z 4 - magc labelng. Hence, B n s Z 4 - magc for all n N. Example.6. Z 4 - magc labelngs of B and B are gven n Fgure 5 and Fgure 6 respectvely. u u u v v v u v Fgure 5: Z 4 - magc labelng of B. u u u v v v Fgure 6: Z 4 - magc labelng of B. Theorem.7. The graph F (t n s Z 4 - magc where t denotes the number of copes of the fan F n. Proof: Let the vertex set and the edge set be gven by V (F (t n E(F n (t = {uv (j Case : Suppose n = 4k and t N. / n, j t} {v (j v (j + = {u, v (j / n, j t} and / n, j t}.

6 66 V.L. Stella Arputha Mary, S. Navaneethakrshnan and A. Nagarajan Defne f : E(F n (t Z 4 {0} by f(uv ( =, j t f(uv n (j ( =, j t f uv (j + ( =, n, j t f v (j = for n, j t. v (j + Then f + : V (F (t n Z 4 s gven by ( f + (u = Σ t j=σ n =f = 0 uv (j ( (4k tmes + mod 4 t = f(uv(j + f(v(j v(j ( + (mod 4 = 0, j t f + (v n (j = 0, j t = + v (j + + f(v(j v(j ( + + (mod 4 = 0, n and j t. We get f + s constant and equals to 0 for all vertces of F n (t. Case : Suppose n = 4k + and t N where k N. Let f : E(F n (t Z 4 {0} be defned as follows: = j t f(uv n (j = j t = n, j t v(j = n, j t v (j + = n, j t. Then f + : V (F n (t Z 4 s gven by f + (u = Σ t j=σ n = ( (4k tmes + t (mod 4 = 0 = f(uv(j + f(v(j v(j ( + (mod 4 = 0, j t f + (v n (j ( + (mod 4 = 0, j t = + v (j + + f(v(j ( + + (mod 4 = 0 v(j Thus, = 0 for n, j t. Hence f + s a constant mappng and s equal to 0

7 Z 4p - Magc labelng for some specal graphs 67 for all vertces n F (t n. Case : Suppose n = 4k. Sub case (: t 0 (mod. Defne f : E(F n (t Z 4 {0} as = = f(uv(j n j t =, n, j t v(j =, n/, j t v(j + =, (n /, j t. Then f + : V (F n (t Z 4 s gven by f + (u = Σ t j=σ n = = ( (4k tmes +.t 0 (mod 4 = 0 = f(uv(j + f(v(j v(j j t = ( + 0 (mod 4 = 0 = + v (j + + f(v(j v(j, n, j t = ( (mod 4, n, j t = 0 n = ( + 0 (mod4, j t Hence f + s a constant mappng and s equal to 0 for all vertces n F n (t. Sub case (: t (mod. Let f : E(F n (t Z 4 {0} be defned as = = f(uv(j n j t =, n, j t v(j =, n/, j t v(j + =, (n /, j t Then f + : V (F n (t Z 4 s gven by f + (u = Σ t j=σ n = ( (4k tmes +.t (mod 4.t = ( + + (mod 4, n, j t = ( + (mod 4 = = n ( + (mod 4, j t. Hence, f + s a constant mappng and s equal to for all vertces n F (t n.

8 68 V.L. Stella Arputha Mary, S. Navaneethakrshnan and A. Nagarajan Case 4: Suppose n = 4k + and t N. Let f : E(F n (t Z 4 {0} be defned as follows: = = f(uv(j n j t =, n, j t v(j =, n/, j t v(j + =, (n /, j t Then f + : V (F n (t Z 4 s gven by f + (u = Σ t j=σ n = f + (u ( k tmes +.t(mod 4 0 (mod 4 = 0 ( + (mod 4 = 0 = f + (v n (j ( + (mod 4, j t ( (mod 4 = 0, n, j t. Hence f + s a constant mappng and s equal to for all vertces n F n (t. In all the cases F n (t admts Z 4 -magc labelng. Hence, F n (t s Z 4 - magc. Example.8. Z 4 -magc labelngs of some one pont unon of fans are gven n ths example. v ( v ( v ( v ( u v ( v ( v ( v ( v ( Fgure 7: Z 4 - magc labelng of F ( v ( v ( v ( v ( 4. u v ( v ( v ( v ( 4 Fgure 8: Z 4 - magc labelng of F ( 4.

9 Z 4p - Magc labelng for some specal graphs 69 v ( v ( v ( v ( 4 v ( v ( u v ( v ( 4 v ( v ( v ( v ( 4 Fgure 9: Z 4 - magc labelng of F ( 4. v ( v ( v ( v ( 4 v ( 5 v ( 6 u v ( v ( v ( v ( 4 v ( 5 v ( 6 Fgure 0: Z 4 - magc labelng of F ( 6. Observaton.9. In all the theorems, f we multply the edge labelng by a postve nteger p, the vertex labelng remans to be a constant and s equal to p tmes the constant value we obtaned. Hence all the above graphs admt Z 4p -magc labelng. Hence, S (P n, T n, B n and F n (t are all Z 4p -magc graphs. References [] S. Amutha and K.M. Kathresan, The exstence and constructon of certan types of labelng for graphs, Ph.D. Thess, Madura Kamaraj Unversty, 006. [] J.A. Galan, A dynamc survey graph labelng, Electronc Journal of Combnatorcs, 7 (00 DS6. [] R.B. Gnanajoth, Topcs n graph theory, Ph.D. Thess, Madura Kamaraj Unvaersty, 99.

10 70 V.L. Stella Arputha Mary, S. Navaneethakrshnan and A. Nagarajan [4] M.C. Kong, S.M. Lee and H.S.H. Sun, On magc strength of graph, Ars Combn., 45, ( [5] R.M. Low and S.M. Lee, On group-magc Euleran graphs, J.Combn. Math. Compn. Comput., 50 ( [6] C. Sekar, Studes n graph theory, Ph.D. Thess, Madura Kamaraj Unversty, 00.