Cordial and 3-Equitable Labeling for Some Star Related Graphs

Transcription

1 Internatonal Mathematcal Forum, 4, 009, no. 31, Cordal and 3-Equtable Labelng for Some Star Related Graphs S. K. Vadya Department of Mathematcs, Saurashtra Unversty Rajkot , Gujarat, Inda samrkvadya@yahoo.co.n N. A. Dan Government Polytechnc, Junagadh , Gujarat, Inda nlesh a d@yahoo.co.n K. K. Kanan Atmya Insttute of Technology and Scence Rajkot , Gujarat, Inda kanankkk@yahoo.co.n P. L. Vhol V V P Engg. College, Rajkot , Gujarat, Inda vholprakash@yahoo.com Abstract We present here cordal and 3-equtable labelng for the graphs obtaned by jonng apex vertces of two stars to a new vertex. We extend these results for k copes of stars. Mathematcs Subject Classfcaton: 05C78 Keywords: Cordal labelng, 3-equtable labelng 1. Introducton We begn wth smple, fnte, connected, undrected graph G =(V,E). In the present work K denote the star. Vertex corresponds to K 1 s called an apex vertex. For all other termnology and notatons we follow Harary[7].

2 1544 S. K. Vadya, N. A. Dan, K. K. Kanan and P. L. Vhol We wll gve bref summary of defntons whch are useful for the present nvestgatons. Defnton 1.1 Consder two stars K (1) and K() then G =< K(1) : K() > s the graph obtaned by jonng apex vertces of stars to a new vertex x. Note that G has n + 3 vertces and n + edges. Defnton 1. Consder k copes of stars namely K,K (1),K (),...K (3). (k) Then the G =< K (1) : K (3) :...: K (k) > s the graph obtaned by jonng apex vertces of each K (p 1) and K (p) to a new vertex x p 1 where p k. Note that G has k(n +) 1 vertces and k(n +) edges. Defnton 1.3 If the vertces of the graph are assgned values subject to certan condtons s known as graph labelng. Most nterestng graph labelng problems have three mportant characterstcs. 1. a set of numbers from whch the labels are chosen.. a rule that assgns a value to each edge. 3. a condton that these values must satsfy. For detal survey on graph labelng one can refer Gallan[6]. Vast amount of lterature s avalable on dfferent types of graph labelng. Accordng to Beneke and Hegde[] graph labelng serves as a fronter between number theory and structure of graphs. Labeled graph have varety of applcatons n codng theory, partcularly for mssle gudance codes, desgn of good radar type codes and convoluton codes wth optmal autocorrelaton propertes. Labeled graph plays vtal role n the study of X-Ray crystallography, communcaton network and to determne optmal crcut layouts. A detal study of varety of applcatons of graph labelng s gven by Bloom and Golomb[3]. Defnton 1.4 Let G =(V,E) be a graph. A mappng f : V (G) {0,1} s called bnary vertex labelng of G and f(v) s called the label of the vertex v of G under f. For an edge e = uv, the nduced edge labelng f : E(G) {0, 1} s gven by f (e)= f(u) f(v). Let v f (0), v f (1) be the number of vertces of G havng labels 0 and 1 respectvely under f and let e f (0),e f (1) be the number of edges havng labels 0 and 1 respectvely under f. Defnton 1.5 A bnary vertex labelng of a graph G s called a cordal

3 Cordal and 3-equtable labelng 1545 labelng f v f (0) v f (1) 1 and e f (0) e f (1) 1. A graph G s cordal f t admts cordal labelng. The concept of cordal labelng was ntroduced by Caht[4]. Many researchers have studed cordalty of graphs. e.g.caht [4] proved that tree s cordal. In the same paper he proved that K n s cordal f and only f n 3. Ho et al.[8] proved that uncyclc graph s cordal unless t s C 4k+. Andar et al.[1] dscussed cordalty of multple shells. Vadya et al.[9],[10],[11] have also dscussed the cordalty of varous graphs. Defnton 1.6 Let G =(V,E) be a graph. A mappng f : V (G) {0, 1, } s called ternary vertex labelng of G and f(v) s called label of the vertex v of G under f. For an edge e = uv, the nduced edge labelng f : E(G) {0, 1, } s gven by f (e) = f(u) f(v). Let v f (0), v f (1), v f () be the number of vertces of G havng labels 0, 1, respectvely under f and e f (0), e f (1), e f () be the number of edges havng labels 0, 1, respectvely under f. Defnton 1.7 A ternary vertex labelng of a graph G s called a 3-equtable labelng f v f () v f (j) 1 and e f () e f (j) 1 for all 0, j. A graph G s 3-equtable f t admts 3-equtable labelng. The concept of 3-equtable labelng was ntroduced by Caht[5]. Many researchers have studed 3-equatablty of graphs. e.g.caht [5] proved that C n s 3-equtable except n 3(mod6). In the same paper he proved that an Euleran graph wth number of edges congruent to 3(mod6) s not 3-equtable. Youssef[1] proved that W n s 3-equtable for all n 4. In the present nvestgatons we prove that graphs <K (1) : K() > and <K (1) : K (3) :...: K (k) > are cordal as well as 3-equtable.. Man Results Theorem-.1: Graph <K (1) > s cordal. Proof: Let v (1) 1,v(1),v(1) 3,...v(1) n be the pendant vertces K (1) and v() 1,v(),v() 3,...v n () be the pendant vertces K. () Let c 1 and c be the apex vertces of K (1) and K () respectvely and they are adjacent to a new common vertex x. Let G =< K (1) >. We defne bnary vertex labelng f : V (G) {0, 1} as follows. For any n N and =1,,...n where N s set of natural numbers. In ths case we defne labelng as follows Case 1: If n even ) = 0; f 1 n =1; n+ n f(c 1 )=0; f(c )=1;

4 1546 S. K. Vadya, N. A. Dan, K. K. Kanan and P. L. Vhol f(x) =0; Case : If n odd ) = 0; f 1 n 1 =1; n+1 n f(c 1 )=f(c )=f(x) =0; The labelng pattern defned above covers all possble arrangement of vertces. The graph G satsfes the condtons v f (0) v f (1) 1 and e f (0) e f (1) 1 as shown n Table 1..e. G admts cordal labelng. Table 1 For better understandng of the above defned labelng pattern, consder followng llustraton. Illustraton. Consder G =< K (1) 1,7 : K () 1,7 >. Here n = 7. The cordal labelng s as shown n Fgure 1. Fgure 1 Above result can be extended for k copes of K as follows. Theorem.3 Graph <K (1) : K (3) :...: K (k) > s cordal. Proof: Let K (j) be k copes of star K, v (j) be the pendant vertces of K (j) (here =1,,...n and j =1,,... k).let and c j be the apex vertex of K (j) x 1,x... x k 1 be the vertces such that c p 1 and c p are adjacent to x p 1 where p k. Consder G =< K (1) : K (3) :... : K (k) >. To defne bnary vertex labelng f : V (G) {0, 1} we consder followng cases. Case 1: n N even and k where k N {1, }. In ths case we defne labelng functon f as,...k ) = 0; f 1 n.

5 Cordal and 3-equtable labelng 1547 =1;f n+ n. f(c j ) = 1; f j even. =0;fj odd. f(x j ) = 1; f j even, j k. =0;fj odd, j k. Case : n N {1, } odd and k where k N {1, }. In ths case we defne labelng functon f as,...k ) = 0; f 1 n 1. =1;f n+1 n. f(c j ) = 1; f j even. =0;fj odd. f(x j )=0,j k. The labelng pattern defned above covers all the possbltes. In each case, the graph G under consderaton satsfes the condtons v f (0) v f (1) 1 and e f (0) e f (1) 1 as shown n Table..e. G admts cordal labelng. Let n =a + b and k =c + d where a N {0},c N Table For better understandng of the above defned labelng pattern, consder followng llustraton. Illustraton.4 Consder G =< K (1) 1,6 : K () 1,6 : K (3) 1,6 >. Here n = 6 and k =3. The cordal labelng s as shown n Fgure. It s the case 1 of Theorem.3. Fgure Theorem.5 Graph <K (1) > s 3-equtable. Proof:Let v (1) 1,v(1),v(1) 3,...v(1) n be the pendant vertces K (1) and v() 1,v(),v() 3,...v () n be the pendant vertces K (). Let c 1 and c be the apex vertces of K (1)

6 1548 S. K. Vadya, N. A. Dan, K. K. Kanan and P. L. Vhol and K () respectvely and they are adjacent to a new common vertex x. Let G =< K (1) : K() >. To defne ternary vertex labelng f : V (G) {0, 1, } we consder the followng cases. Case 1: n 0(mod3) In ths case we defne labelng f as )=0; 0(mod3) =1; 1(mod3) =; (mod3), 1 n 1 f(v n (1) )=1; f(v n () )=f(c 1 )=f(x) =0; f(c )=; Case : n 1(mod3) In ths case we defne labelng f as: )=0; 0(mod3) =1; 1(mod3) =; (mod3) f(c 1 )=f(x) =0; f(c )=; Case 3: n (mod3) In ths case we defne labelng f as )=0; 0(mod3) =1; 1(mod3) =; (mod3) f(c 1 )=f(c )=f(x) =0; The labelng pattern defned above covers all possble arrangement of vertces. In each case, the graph G under consderaton satsfes the condtons v f () v f (j) 1 and e f () e f (j) 1 for all 0, j as shown n Table 3..e. G admts 3-equtable labelng. Let n =3a + b and a N {0} Table 3 For better understandng of the above defned labelng pattern, consder followng llustraton. Illustraton.6 Consder a graph G =< K (1) 1,8 : K() 1,8 > Here n = 8.e n (mod3). The correspondng 3-equtable labelng s shown n Fgure 3. It

7 Cordal and 3-equtable labelng 1549 s the case related to case -3 Fgure 3 Above result can be extended for k copes of K as follows. Theorem.7 Graph <K (1) : K() : K(3) :...: K(k) > s 3-equtable. Proof: Let K (j), j =1,,...kbe k copes of star K. Let v (j) be the pendant vertces of K (j) where =1,,...n and j =1,,...k. Let c j be the apex vertex of K (j) where j =1,,...k. Let G =< K(1) : K() : K(3) :...: K(k) > and x 1,x,...,x k 1 are the vertces as stated n Theorem.3. To defne ternary vertex labelng f : V (G) {0, 1, } we consder followng cases. Case 1: For n 0(mod3) In ths case we defne labelng functon f as follows Subcase 1: For k 0(mod3) ) = 0; f 1(mod3) =1;f (mod3) =;f 0(mod3), n 1 f(v n (j) ) = 1; f j 1, (mod3) =;fj 0(mod3) f(c j ) = 0; f j 1, (mod3) =;fj 0(mod3) f(x j ) = ; f j n 1 Subcase : For k 1(mod3) f(v (1) ) = 0; f 1(mod3) =1;f (mod3) =;f 0(mod3) f(c 1 )= f(x 1 )=0 For remanng vertces take j = k 1 and use the pattern of subcase 1. Subcase 3: For k (mod3) ) = 0; f 1(mod3)

8 1550 S. K. Vadya, N. A. Dan, K. K. Kanan and P. L. Vhol =1;f (mod3) =;f 0(mod3), 1 n 1 f(v n (1) )=1 f(v n () )=f(c )=f(x j )= f(c 1 )=0 For remanng vertces take j = k and use the pattern of subcase 1. Case : For n 1(mod3) In ths case we defne labelng functon f as follows Subcase 1: For k 0(mod3) Subcase 1.1: For n =1 1 ) = ; f j 0(mod3) =1;fj 1, (mod3) f(c j ) = ; f j 1(mod3) =1;fj (mod3) =0;fj 0(mod3) f(x j )=0;j k Subcase 1.: For n>1 ) = 0; f 0(mod3) =1;f 1(mod3) =;f (mod3), n n 1 ) = 0; f j 1, (mod3) =;fj 0(mod3) f(v n (j) )=1 f(c j ) = ; f j 1(mod3) =0;fj 0, (mod3) f(x j ) = 0; f j 1, (mod3) =;fj 0(mod3), j k Subcase : For k 1(mod3) f(v (1) ) = 0; f 0(mod3) =1;f 1(mod3) =;f (mod3) f(c 1 )=0 f(x 1 )= For remanng vertces take j = k 1 and use the pattern of subcase 1.1 or subcase 1. fn = 1orn > 1 respectvely. Subcase 3: For k (mod3). ) = 0; f 0(mod3) =1;f 1(mod3) =;f (mod3) f(c 1 )=f(x )=

9 Cordal and 3-equtable labelng 1551 f(c )=f(x 1 )=0 f(x 1 ) = ; f n =1 f(x 1 ) = 0; f n>1 For remanng vertces take j = k and use the pattern of subcase 1.1 or subcase 1. fn = 1orn > 1 respectvely. Case 3: For n (mod3). In ths case we defne labelng functon f as follows Subcase 1: For k 0(mod3) ) = 0; f 0(mod3) =1;f 1(mod3) =;f (mod3), n 1 ) = 1; f j 1(mod3) =;fj 0, (mod3) f(c j ) = ; f j 1(mod3) =0;fj 0, (mod3) f(x j ) = 0; f j 1, (mod3) =;fj 0(mod3) n Subcase : For k 1(mod3) f(v (1) ) = 0; f 0(mod3) =1;f 1(mod3) =;f (mod3), n f(c 1 )=0 f(x 1 )= For remanng vertces take j = k 1 and use the pattern of subcase 1. Subcase 3: For k (mod3) ) = 0; f 0(mod3) =1;f 1(mod3) =;f (mod3), n f(c 1 )=. f(c )=f(x j )=0. For remanng vertces take j = k and use the pattern of subcase 1. The labelng pattern defned above covers all possble arrangement of vertces. In each case, the graph G under consderaton satsfes the condtons v f () v f (j) 1 and e f () e f (j) 1 for all 0, j as shown n Table 4..e. G admts 3-equtable labelng. Let n =3a + b and k =3c + d where a N {0},c N.

10 155 S. K. Vadya, N. A. Dan, K. K. Kanan and P. L. Vhol Table 4 For better understandng of the above defned labelng pattern, consder followng llustraton. Illustraton.8 Consder a graph G =< K (1) 1,5 : K () 1,5 : K (3) 1,5 : K (4) 1,5 >. Here n = 5 and k = 4. The correspondng 3-equtable labelng s as shown n Fgure 4. Fgure 4 3. Concludng Remarks Labeled graph s the topc of current nterest for many researchers as t has dversfed applcatons. We dscuss here cordal labelng and 3-equtable labelng of some star related graphs. Ths approach s novel and contrbute two new graphs to the theory of cordal graphs as well as 3-equtable graphs. The derved labelng pattern s demonstrated by means of elegant llustratons whch provdes better understandng of the derved results. The results reported here are new and wll add new dmenson n the theory of cordal and 3-equtable graphs. References [1] M Andar, S Boxwala and N B Lmaye: A Note on cordal labelng of multple shells, Trends Math. (00), [] L W Beneke and S M Hegde,Strongly Multplcatve graphs,dscuss.math. Graph Theory,1(001), [3] G S Bloom and S W Golomb, Applcatons of numbered undrected graphs, Proceedngs of IEEE, 165(4)(1977),

11 Cordal and 3-equtable labelng 1553 [4] I Caht, Cordal Graphs: A weaker verson of graceful and harmonous Graphs, Ars Combnatora, 3(1987), [5] I Caht, On cordal and 3-equtable labelngs of graphs, Utl. Math., 37(1990), [6] J A Gallan, A dynamc survey of graph labelng, The Electroncs Journal of Combnatorcs, 16(009) DS6. [7] F Harary, Graph theory, Addson Wesley, Readng, Massachusetts, 197. [8] Y S Ho, S M Lee and S C Shee, Cordal labelng of uncyclc graphs and generalzed Petersen graphs, Congress. Numer.,68(1989) [9] S K Vadya, G V Ghodasara, Sweta Srvastav, V J Kanera, Cordal labelng for two cycle related graphs, The Mathematcs Student, J. of Indan Mathematcal Socety, 76(007) [10] S K Vadya, G V Ghodasara, Sweta Srvastav, V J Kanera, Some new cordal graphs, Int. J. of scentfc copm.,(1)(008) [11] S K Vadya, Sweta Srvastav, G V Ghodasara, V J Kanera, Cordal labelng for cycle wth one chord and ts related graphs. Indan J. of Math. and Math.Sc 4() (008) [1] M. Z. Youssef, A necessary condton on k-equtable labelngs, Utl. Math., 64 (003) Receved: November, 008