SUPER GRACEFUL LABELING FOR SOME SPECIAL GRAPHS

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1 IJRRAS 9 ) Deceber pdf SUPER GRACEFUL LABELING FOR SOME SPECIAL GRAPHS M.A. Perual, S. Navaeethakrsha, S. Arockara & A. Nagaraa 4 Departet of Matheatcs, Natoal Egeerg College, K.R.Nagar, Kovlpatt, Tal Nadu, Ida.,4 Departet of Matheatcs, V.O.C College, Thoothukud, Tal Nadu, Ida. Departet of Matheatcs, Mepco Schlek Egeerg College, Svakas, Tal Nadu, Ida Eal : eetperual.a@gal.co, sk.voc@gal.co, sarockara_77@yahoo.co, 4 agaraa.voc@gal.co ABSTRACT Let G be a p,q) graph. A bectve fucto f:vg) U EG) {,,...,p+q} such that fuv) fu)-fv) for every edge uv єeg) s sad to be a super graceful labelg. A graph G s called a super graceful graph f t adts a super graceful labelg. I ths paper, we show that the graphs P -,,...,), Cocout tree, K,, S, ad B,,k) are super graceful graphs. Keywords: Graceful labelg, Super graceful labelg ad Super graceful graphs.. INTRODUCTION By a graph, we ea a fte udrected graph wthout loops or ultple edges. A path of legth s deoted by P. A cycle of legth s deoted by C. G + s a graph obtaed fro the graph G by attachg a pedet vertex to each vertex of G. The cocept of graceful labelg has bee troduced by Rosa [] 967. A fucto f s a graceful labelg of a graph G wth p vertces ad q edges f f s a ecto fro the vertces of G to the set {,,...,q} such that whe each edge uv s assged the label fu) fv),the resultg edge labels are dstct. The gracefuless of graphs otvates us to defe a ew type of labelg, called Super graceful labelg [6]. Let G be a p, q) - graph. A bectve fucto f : V G) EG) {,,..., p + q} such that fuv) fu) fv) for every edge uv є EG) s sad to be a super graceful labelg. A graph G s called a super graceful graph f t adts a super graceful labelg. I ths paper, we show that the graphs P,,..., ), Cocout tree, K,, S, ad B,, k) are super graceful graphs.. MAIN RESULTS Defto.. [] The graph P,,,..., ) s a graph obtaed fro a path of vertces v, v,..., v havg path legth by og pedet vertces at each of th vertex. The pedet vertces are labeled as u, ; u, ;...; u, for. Theore.. P,,..., ) s a super graceful graph, for. Proof. Let G P,,..., ). Now VG) ++)/, EG)-+ +)/ ad V G) U EG) -++) +-. Defe f : V G) EG) {,,..., + } as follows ),, od ) ) f v ),, 0 od ) f u, ) ad od ) For f u, ) ), For ad 0 od ) ) f u, ), 8

2 IJRRAS 9 ) Deceber 0 Perual & al. Super Graceful Labelg for Soe Specal Graphs We costruct the vertex labeled sets as follows: Let V od ) { f v od ) ) or) V, 7, 7,..., 4 6, 7, 7,..., ) accordg as s odd or eve. 0 od ) { f v ) 0 od ) {,,,..., accordg as s odd or eve. V { f {} V u, } or),,,..., { f 0 od ) 4 u, 0 od ) 8

3 IJRRAS 9 ) Deceber 0 Perual & al. Super Graceful Labelg for Soe Specal Graphs 84 : : 7 : or : : 7 : ) accordg as s odd or eve. ad { ), 5 u f od V od ) ) ) :... 5} : { } : { } : { ) or :... 5} : { } : { } : { accordg as s odd or eve. 4 4,...,,... {,5,7,9,} {5,7,9} {} or) 6 4,...,,... {,5,7,9,} {5,7,9} {} accordg as s odd or eve. 4,..., 5, ;5,7,9;,5,7,9,;...; or) 6,..., 6, ;5,7,9;,5,7,9,;..., accordg as s odd or eve We costruct the edge labeled sets as follows: } { ) v v f od E Let

4 IJRRAS 9 ) Deceber 0 Perual & al. Super Graceful Labelg for Soe Specal Graphs { f v ) f v ) } od ) od ) ) ) ) od ) od ) { 6 4) ) ) ) } { or) { E 4, 4, 8,...,4 } 8,..., } accordg as s odd or eve. { f v v 0 od ) { f v ) f v ) } 0 od ) ) 0 od ) ) 85

5 IJRRAS 9 ) Deceber 0 Perual & al. Super Graceful Labelg for Soe Specal Graphs ) 0 od ) } ) { ) 0 od ) } ) { ) 0 od } 54,..., 8, 0, { or) } 54,...,4 8, 0, { accordg as s odd or eve., { ) v u f od E ) } ) { ), u f v f od }) ) ) ) { ) od od ) )

6 IJRRAS 9 ) Deceber 0 Perual & al. Super Graceful Labelg for Soe Specal Graphs 87 ) } { ) od,...,}, {... 8},..., 0, { 6} 4,, { } { ) or 4},...,,4 {4... 6} 4,, { } { accordg as s odd or eve. ) ) 0, 4 v u f od E ) ) ) 0, u f v f od od ) ) 0 ) 0 od ) ) ) ) 0 od ) ) ) ) 0 od

7 IJRRAS 9 ) Deceber 0 Perual & al. Super Graceful Labelg for Soe Specal Graphs { { 6, 6, 8} { or) 0, 8} {... {,,..., 4,...,}, 0, 4, 6},... {4,4,4 4,..., 4} 4, 6} accordg as s odd or eve. Fro the vertex labeled sets ad the edge labeled sets, we observe that they are dstct. Ther uo s {,,..., }. Therefore, f s a super graceful labelg ad hece, P,,..., ) s a super graceful graph. Exaple. The graphs ) show Fg. ad Fg. P 4,,,4,5 ad,,,4,5 5,6) P adt super graceful labelg, such as those Fg. Fg. Defto.4. [] A graph G V, E ) s called bpartte f V V V wth V V, ad every edge of G s of the for u, v wth u V ad v V. If each vertex V s oed wth every vertex V, we have a coplete bpartte graph. I ths case V ad V, the graph s deoted by K,. Theore.5. Every coplete bpartte graph, ) s super graceful. K, V V V where { u, u,..., u} { v, v,..., v K ) E K, ) : V K, ) E K, ) {,,..., as follows. Proof. Let Now, V, ad V ad } Defe f } f u ), ad f v ) ),. We costruct the vertex labeled sets as follows: V. 88

8 IJRRAS 9 ) Deceber 0 Perual & al. Super Graceful Labelg for Soe Specal Graphs Let ad V { f u { } {,,..., } V { f v { ) } { ), ),..., We costruct the edge labeled set as follows: Let E { f u v { f u ) f v ) } { ) ) } { ) } { ), ),..., ) } { ), ),..., ) } { ), ),..., ) }... { ), ),..., ) } { ), ),..., ) } { ), ),..., ) }... {, ),...,. We observe that all the vertex labeled sets ad the edge labeled set are dstct ad ther uo s {,,..., }. Therefore, f s a super graceful labelg ad hece, K,, ) s a super graceful graph. Corollary.6 By takg, the proof of the above theore, we get a star graph K, ad t s a super graceful graph. 89

9 IJRRAS 9 ) Deceber 0 Perual & al. Super Graceful Labelg for Soe Specal Graphs Exaple.7 The graph K 4,5 adts super graceful labelg, such as those show Fg.. Theore.8 Cocout tree s a super graceful graph. Proof. Let v,..., Fg. 0, v, v v be the vertces of a path, havg path legth, ) pedet vertces, beg adacet wth v 0. Now, V G) ad E G). Defe f: VG) U EG) {,,...,+ }as follows. v v,..., v, ad be the We costruct the vertex labeled sets as follows:. Let V 0 0 od ) { f v 0 { } 0 od ) {,,..., ) } or){,,..., } accordg as s odd or eve. {,,..., } or) {,,..., } accordg as s odd or eve. accordg as s odd or eve. V 0 {,,,..., } or) {,,..., } accordg as s odd or eve. ad V { f vk {k } {, 5,..., } k k We costruct the edge labeled sets as follows: 90

10 IJRRAS 9 ) Deceber 0 Perual & al. Super Graceful Labelg for Soe Specal Graphs E 0 { f v v 0 od ) 0 0 accordg as s odd or eve. {, ), 4),...,} or) {, ), 4),...,4} accordg as s odd or eve. E 0 { f v v od ) 0 { f v ) f v ) } od ) 0 { ) ) } od ) { } 0 { od ) { ), ),..., ) or){ ), ),..., ) accordg as s odd or eve. { ), ),...,4} or) { ), ),...,} accordg as s odd or eve. ad E { f vkv k 0 9

11 IJRRAS 9 ) Deceber 0 Perual & al. Super Graceful Labelg for Soe Specal Graphs { f vk ) f v ) } k 0 { k } k { } k k { ), ),...,} We observe that all the vertex labeled sets are havg odd values ad the edge labeled sets are havg eve values ad they dstct. Ther uo s {,,..., }. Therefore, f s a super graceful labelg ad hece, cocout tree graph s super graceful. Corollary.9 By takg, the path P of the above proof of theore, K, s a super graceful graph. Exaple.0 The cocout tree graphs adt super graceful labelg, such as those show Fg.4 ad Fg.5. Fg.4 Fg.5 9

12 IJRRAS 9 ) Deceber 0 Perual & al. Super Graceful Labelg for Soe Specal Graphs Theore. The Proof. The S, graph s super graceful, for ad. S, graph s obtaed paths, u, u, u..., u ; u u u... u ; u u u,... u ;... ; u u u... u ad detfy the vertces wth u 0. Now, V S ) E S ) ),, f : V S, ) E S, ) {,,..., Defe } as follows: 0 f u ) f u0). For ad od ) f u ) ), ad For ad 0 od ) f u ) ), 0 Let V { f u { f u0 {} u 0 0 0, u,..., u V { f u od ) { ) } od ) { ), ),..., ) } or) { ), ),..., ) } accordg as s odd or eve. {,,..., ;,,..., ;...;,,..., } or) {,,..., ;,,..., ;...;,,..., } accordg as s odd or eve. 9

13 IJRRAS 9 ) Deceber 0 Perual & al. Super Graceful Labelg for Soe Specal Graphs V { f u 0 od ) { ) } 0 od ) { ), ) 4,..., ) } or) { ), ) 4,..., ) } accordg as s odd or eve. {,,..., ;,,..., ;,,..., } or) {,,..., ;,,..., ;,,..., } accordg as s odd or eve. We costruct the edge labeled sets as follows: 0 Let E { f u u { f u ) f 0 u ) } { ) ) ) } { ) } 94

14 IJRRAS 9 ) Deceber 0 Perual & al. Super Graceful Labelg for Soe Specal Graphs { ) {, ),...,} E u f u od ) f u f u od ) { ) ) ) )) } od ) od ) { 4 ) } { 4 ) } od ) {, 4,8,..., 4 ) } od ) { ), ),..., ) ) } od ) { ), ), 5),..., { ), ),...,... { ), ),..., 95

15 IJRRAS 9 ) Deceber 0 Perual & al. Super Graceful Labelg for Soe Specal Graphs or) { ), ),..., { ), ),...,... { ), ),..., accordg as s odd or eve. E f u u 0 od ) f 0 od ) u f u ) ) ) ) 0 od ) 0 od ) 4 ) { ), ),..., 0 od ) { ), 4), 6),..., { ), 4), 6),...,... { ), 4),..., or) { ), 4), 6),..., { ), 4),...,... { ), 4),..., accordg as s odd or eve. I both the cases s odd or eve), we observe that all the vertex labeled sets are havg odd values ad the edge labeled sets are havg eve values ad they are dstct. Ther uo s {,,..., }. 96

16 IJRRAS 9 ) Deceber 0 Perual & al. Super Graceful Labelg for Soe Specal Graphs Therefore, f s a super graceful labelg ad hece, S,, ) s a super graceful graph. Corollary. By takg S, graph, we get S, ad t s a super graceful graph. By takg S, we get a path graph ad t s a super graceful graph. ad Exaple. The graphs S 7,6 ad S 7,9 adt super graceful labelg, such as those show Fg.6 ad Fg. 7. Fg. 6 Fg.7 Defto.4 The graph B,, k) s a graph obtaed fro a path of legth k by attachg the star ad K, wth ts pedet vertces. Theore.5 The graph B,, k) s a super graceful graph. Proof. Let u,, u,,..., u, be adacet vertces to v 0 ad u,, u,,..., u, be aother set of adacet vertces to v k. Let v 0 ad v k be teral vertces of a path P... k v0vv vk. Let G B,, k) Now, V G) k, E G) k Case k s odd. K, 97

17 IJRRAS 9 ) Deceber 0 Perual & al. Super Graceful Labelg for Soe Specal Graphs Defe f : V G) E G) {,,..., k) } as follows. k), 0 k, 0 od ) f u, ),. f v ), 0 k, od ) f u, l ) k l, l. We costruct the vertex labeled sets as follows: Let V { f u, { } {,,5,..., } k V 0 0 od ) { f v k 0 { k) } 0 od ) { k), k),..., ) k} V k 0 od ) { f v k 0 { } od ) {,,..., k} ad V4 { f u, l l 98

18 IJRRAS 9 ) Deceber 0 Perual & al. Super Graceful Labelg for Soe Specal Graphs l { l k} { k, k 4,..., k } We costruct the edge labeled sets as follows: Let E { f v0u, { f v0) f u, ) } { k) ) } { k), k),..., k) ) } { k), k),..., k) } k E 0 { f v v 0 od ) k 0 { f v ) f v ) } 0 od ) k 0 { k) ) ) } 0 od ) k 0 { k ) } 0 od ) { k), k ), k 4),..., 99

19 IJRRAS 9 ) Deceber 0 Perual & al. Super Graceful Labelg for Soe Specal Graphs k E { f v v od ) k { f v ) f v ) } od ) k { ) k) )) } od ) k { k)) } od ) k { k od ) { k ), k ),..., E4 { l f v u k l { f vk ) f u,l ) } l { ) k ) ) k l) } l {l} l {,4,..., } 400

20 IJRRAS 9 ) Deceber 0 Perual & al. Super Graceful Labelg for Soe Specal Graphs Case k s eve Defe f : V G) E G) {,,..., k) } as follows. k), 0 k, 0 od ) f u, ), f v ), 0 k, od ) f u, l ) ) k l, l. We costruct the vertex labeled sets as follows: ' Let V { f u, 0 { } 0 {,,..., } V k 0 0 od ) { f v ' k 0 { k) } 0 od ) { k), k),..., ) k } k V 0 od ) { f v ' k 0 { } od ) {,,..., k } ad 40

21 IJRRAS 9 ) Deceber 0 Perual & al. Super Graceful Labelg for Soe Specal Graphs ' V4 { f u, l l { ) k l} l { ) k, ) k,..., k } We costruct the edge labeled sets as follows: ' E { f v0u, { f v0) f u, ) } { k) ) ) } { k ) } { k { k), k ),..., k k ' E { 0 v v f 0 od ) k { f v ) f v ) } 0 0 od ) 40

22 IJRRAS 9 ) Deceber 0 Perual & al. Super Graceful Labelg for Soe Specal Graphs k { k) ) ) } 0 0 od ) k { k 0 0 od ) { k), k ),..., k ' E { f v v od ) k { f v ) f v ) } od ) k { ) k) )) } od ) k { k)) } od ) k { k od ) { k ), k ),..., ad ' E4 {, l f v u k l { f vk ) f u,l ) } l 40

23 IJRRAS 9 ) Deceber 0 Perual & al. Super Graceful Labelg for Soe Specal Graphs { ) k ) ) k l) } l { l} {,4,..., } l I both the cases, we observe that all the vertex labeled sets are havg odd values ad the edge labeled sets are havg eve values ad they are dstct. Ther uo s {,,..., k) }. Therefore, f s a super graceful labelg ad hece, B,, k) s a super graceful graph. Corollary.6 By takg k, the proof of the above theore, B, ) s a super graceful graph. Exaple.7 The graphs B 8,9,5) ad B 9,8,4) adt super graceful labelg, such as show Fg.8 ad Fg.9. Fg.8 Fg. 9 REFERENCES []. Davd M.Burto, Eleetary Nuber Theory, Sxth Edto, Tata McGraw - Hll Edto, Teth reprt, 00. []. G.A.Galla, A Dyac Survey of Graph Labelg, The Electroc Joural of Cobatorcs 6009) # DS 6, pp 9. []. A.Rosa, O certa valuatos of the vertces of a graph, Theory of graphs Iteratoal Syposu, Roe, 966. [4]. K.M.Kathresa ad S.Autha, Fboacc graceful graphs, Ph.D., Thess, Madura Kaara Uversty, October 006. [5]. M.A.Perual, S.Navaeethakrsha ad A.Nagaraa, Lucas Graceful Labelg for Soe Graphs Iteratoal Joural of Matheatcal Cobatorcs. March 0. Vol., pp. -9. [6]. M.A.Perual, S.Navaeethakrsha, A.Nagaraa ad S.Arockara, Super Graceful Labelg for Soe Sple Graphs - Coucated. 404

V. Hemalatha, V. Mohana Selvi,

V. Hemalatha, V. Mohana Selvi, Iteratoal Joural of Scetfc & Egeerg Research, Volue 6, Issue, Noveber-0 ISSN - SUPER GEOMETRIC MEAN LABELING OF SOME CYCLE RELATED GRAPHS V Healatha, V Mohaa Selv, ABSTRACT-Let G be a graph wth p vertces