Some Simple Algorithms for Some Odd Graceful Labeling Graphs

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1 Proceedng of the 9th WSEAS Internatona Conference on APPLIED INFORMATICS AND COMMUNICATIONS (AIC '09) Some Smpe Agorthm for Some Odd Gracefu Labeng Graph M. Ibrahm Moua Facuty of Computer & Informaton, Benha Unverty, Benha, Egypt, ABSTRACT The pder graph obtaned by dentfyng the end pont of m nternay dont path each of ength n P the graph obtaned by dentfyng the denoted, n other end pont of the graph Pn caed the coed pder graph denoted n. Th paper decrbe the agorthm to how that the foowng graph; P 2r + ; m, r 5 m 2, the graph obtaned by onng one or two path P m to each vertex of the path P n, the graph obtaned by addng two path each of ength m to each vertex of the path p, the coed pder graph, the tenor product of n a two path graph P n P m are odd gracefu. KEY WORDS: Vertex abeng, edge abeng, odd gracefu.. Introducton. The gracefu abeng one of the mot wdey ued; t ha a wde rangng appcaton uch a radar, communcaton network, crcut degn, codng theory, atronomy, x-ray crytaography. The tudy of gracefu graph gracefu abeng method wa ntroduced by Roa []. Roa defned a β- vauaton of a graph G wth q edge an necton from the vertce of G to the et {0,, 2 q} uch that when each edge xy agned the abe f(x)-f(y), the reutng edge are dtnct. β- Vauaton a functon that produce gracefu abeng. However, the term gracefu abeng wa not ued unt Goomb tuded uch abeng evera year ater [2]. A graph G of ze q odd-gracefu, f there an necton f from V(G) to {0,, 2 2q-} uch that, when each edge xy agned the abe or weght f(x)-f(y), the reutng edge abe are {, 3, 5,,2q-}. Th defnton wa ntroduced by Gnanaoth [4] n 99. In 99, Gnanaoth [4] proved the graph Cn K2 odd-gracefu f ony f n even. She ao proved that the graph obtaned from Pn P2 by deetng an edge that on to end pont of the Pn path th at graph knew a the adder graph. In 2000, Kathrean [6] ue the notaton Pn to denote the graph obtaned by dentfyng the end pont of m path each one ha ength n. In 997 Ederg [5] generazed Gnanaoth [4] reut on tar by howng that the graph obtaned by onng one end pont from each of any odd number of path of equa ength odd gracefu graph. In 2002 Sekar[7] proved that the graph; Pn when n 2 m odd, P2 m 2, P4 m 2, P n when n m are even P, P, n 4 m 4, 4r+ ;4r+ 2 4r ;4r a n- poygona nake wth n even are odd gracefu. Sekar tude have produced etmate of P n, but there t ome graph n the form of P n dd not proved to be odd gracefu up to date, th graph are P 3 m 2, P4 r+ ;4r, P4 r ;4r 2, P4 r+ ;4r 2 P4r ;4r+ 2, r. In addton, no agorthmc reearch ha been found that proved the odd gracefu of the above graph. In the current tudy, we preent the methodooge to prove for a m 2the foowng graph P4, r r =,2,3 P 4 r +, r =,2 are odd gracefu. We preent agorthm whch how that the coed pder graph the graph obtaned by onng one or two P path m to each vertex of the path Pn are odd gracefu. We ue the cyce repreentaton denoted - repreentaton to preent a mpe abeng graph agorthm, the cyce repreentaton mar to theπ -repreentaton made by Kotazg [3].. π -repreentaton Let 0, 2 be three dtnct parae ne yng n a pane uch that 2 are both equdtant from 0. Let 2 be nfnte et of pont on 2 repectvey uch that the pont n both 2 are paced eveny at dtance apart. Conder the et of a traght ne between pont of pont of 2 ; et 0 be the et of pont of nterecton of thee ne wth 0. Ceary the pont of 0 are paced at the dtance of /2apart. Kotazg [3] defneπ - repreentaton a a foowng; conder a bpartte graph G parttoned nto vertex et V V2 drawn o that the vertce e on dtnct conecutve pont of 2 repectvey the edge of G nterect dtnct conecutve pont of 0. A graph G ha a π - ISSN: ISBN:

2 Proceedng of the 9th WSEAS Internatona Conference on APPLIED INFORMATICS AND COMMUNICATIONS (AIC '09) repreentaton f t can be drawn n the way hown n fgure..2 -repreentaton Spder graph the graph whch conted of one centra pont connected wth m number of path Pn of the ame ength denoted P nm ;. We defne -repreentaton a the foowng; conder a centra pontv0 urrounded by n cycec m, o that v0 the common center for a the cyce. Let v be the nterecton pont between the path P the cycec where =, 2... m =, 2, 3...n-. The 2 n vertex et of the path P V = { v0} { v, v,..., v }. The -repreentaton of the graph Pnm ; howng n fgure 2 2. Man Reut about the Deveopment of Agorthm wth Repect to Labeng of Pn graph. We are ntroducng an eay method for abeng the vertce of the Pn graph a an odd gracefu abeng. We defne the functon β: {, 2, 3, 4...} {, 3, 5, 7...}. The expreon β( ) = 2- decrbe the functon β of a varabe, whch an nteger, then th functon reate each nput,, wth a nge output, The graph P4r, r =,2,3 (See fgure 3). Put the graph P 4r n the -repreentaton whch cont of the centra pont v0 two cycecc, 2...C 4r Number the centra pont v0 by the functon f ( v 0 ) = If r =,2; Number the vertce on the odd cycec, =, 3, 7 conecutvey n revere the cock a f ( v ) = ((8 r {4, 6,0} ) m β() 4. If r =3; Number the vertce on the odd cyce CCCCC,,,, conecutvey n revere the cock a f ( v ) 20m f ( v7 ) 2m f ( v ) 2m f ( v ) 4m = β(), 3 = β(), 5 = β(), = β(), f ( v9 ) = 8m β() repectvey. 5. Number the vertce on the even cycec conecutvey f ( v ) = 2 m 2 β. n revere the cock a ( ) 2.2 The graph P4r +, r =,2 (See fgure 4). Put the graph P 4r + n the -repreentaton whch cont of the centra pont v0 two cycec, C 2...C4 r. v by the functon ( ) 2. Number the centra pont 0 f v 0 = Number the vertce on the odd cycec =, 3, 5, 7 conecutvey n revere the cock a f ( v ) = ( (8 r {0, 2,8,4} ) m β (). 4. Number the vertce on the even cycec conecutvey f ( v ) = 2 m 2 β n revere the cock a ( ) 3-Coed pder graph The graph n obtaned from pder graph by connectng end pont of a path to one vertex. The vertex et of the coed pder graph n ha three part, the mot eft vertex w, the mot rght vertexu, the ntermedate vertce n the path P... P n, the tota number of edge q= ( n ) m. The -repreentaton of the graph n drawn n the way hown n fgure 5. 3.Agorthm for odd gracefu abeng of n. Put the graph n n the -repreentaton whch cont of the two end pontu abeed by f ( u ), w abeed by f ( w ) the cyce C... Cn 2, we number the cyce from rght to eft. The cycec ha the vertce v abeed by f ( v ), where =, 2 m ; =, 2 n 2, m the number of path each of ength n. 2. Number the mot rght vertex u wth the vaue f ( u) = 0 3. Number the vertce on the odd cycec, C3, C 5,... conecutvey n revere the cock a (2n ) m 2 +, number the vertce on the even cycec2, C4,C 6,... conecutvey n revere the cock a ( + 2) m Compute the tota number of edge q= ( n ) m. If the number of edge odd, we number the vertex w wth vaue f ( w) = ( n 2) m+, otherwe number the vertex w wth vaue f ( w) = ( n+ ) m. ISSN: ISBN:

3 Proceedng of the 9th WSEAS Internatona Conference on APPLIED INFORMATICS AND COMMUNICATIONS (AIC '09) 4-Agorthm for odd gracefu of the graph Pn Pm In fgure 6 (A), we gve an exampe of the graph P n P m, fgure 6 (B) how the path track we foow to abe the graph. We tart abeng from the mot outh eatern vertex of a path P m then we move up to the mot north eatern vertex of path Pn then we move to the wet for one edge then move down through path P m we move to wet to the bottom vertex of next path of Pm then we move up repeat the ame proce unt we reach the mot north (outh) wetern vertex of path P n. In the end the graph appear a a path whch ha vertex et v abeed by f ( v ) ( ) odd f ( v 2q + even 5- The graph ( Pn K2 ) Pm odd gracefu Th graph obtaned by addng two eave to each vertex of path P n then add a path P m to each eaf, ee fgure 7. Th graph odd gracefu abeed by the ame way a Pn Pm (ee fgure 6-B to ee the path track of abeng proce). We tart abeng from the mot outh eatern vertex of a path P m then we move up to the mot north eatern vertex of path P n then we move down to the wet eaf vertex the mot north eatern vertex contnue down move through path P m we move to wet to the bottom vertex of next path of P then we move up repeat m the ame proce unt we reach the mot outh wetern vertex of path P n ee fgure 7. For each vertex v n the vertex et of the graph ha the abeed functon f ( v ) defned a ( ) odd f ( v 2q + even 6. The tenor product of two path are odd gracefu The Tenor product of two graph G G 2 denoted G G 2 vertex et denoted V G G2 = V G ) V G2 uu, v= v, v n ( ) ( ( ) Conder any two pont u= ( ), ( ) V ( G G2) 2 2 the edge et denoted EG ( G2) where EG ( G) = {( u, v)( u, v) : uu EG ( ) vv EG ( )} We are ntroducng an agorthm for abeng the vertce of the Pn Pm graph a an odd gracefu abeng; the tenor ha a tota number q= 2( n )( m ) of edge. graph Pn Pm At the begnnng, we dvded the graph Pn Pm ntom eve, 2,..., m. An arbtrary eve ha a vertex etv for a =, 2,...n =, 2,..., m whch are abeed odd gracefuy a ( ) f v ee fgure 8. Agorthm for Odd Gracefu Labeng of the graph P n P m. - For each vertex v n the frt eve number the vertex a f ( v ) =, =,2,..., n 2- If m even 2. For each eve, odd more than one number t vertce ( = ; n; += 2 ) v accordng the for oop For f( v ) = ( + )( n ) + ( ) For ( = 2 ; n; += 2 ) f ( v ) = 4( m 2 )( n ) ( ( ) n ( + ) ) 2.2 For each eve, even more than or equa two number t vertcev accordng the for oop For ( = ; n; += 2 ) f( v ) = ( 2)( n ) + For n 3- Ee( m odd) 3. If m = 3 ( = 2 ; ; += 2 ) f ( v ) = 4( m 2 )( n ) ( ( 2) n ( ) ) 3 (4n 5) + odd fv ( 2 n even m > ( 3) ( 4) odd m m+ n m+ + fv ( 4 m 2 ( n ) (( m 3) n ( m )) even 3.2 If 3 7. Concuon In th work we condered many type of odd gracefu abeng of pder, coed pder graph, the graph obtaned by onng one or two path P m to each vertex of the ISSN: ISBN:

4 Proceedng of the 9th WSEAS Internatona Conference on APPLIED INFORMATICS AND COMMUNICATIONS (AIC '09) path P n, the graph obtaned by tenor product of a two path graph P n P m. We defne a new graph repreentaton named repreentaton durng our decrpton of the odd gracefu abeng agorthm. Further work need to be done to etabh whether the graph P4 r+ ;4r, P4 r ;4r 2, P4 r+ ;4r 2 P4r ;4r 2, + r 3 are odd gracefu. Reference: [] A. Roa (967), On certan vauaton of the vertce of a graph, Theory of Graph (Internat. Sympoum, Rome, Juy 966), Gordon Breach, N. Y. Dunod Par [2] S. W. Goomb (972), How to number a graph: graph theory computng, R. C. Read, ed., Academc Pre: [3] A. Kotazg, On certan vertex vauaton of fnte graph, Utta Math.4 (973) [4] R.B. Gnanaoth (99), Topc n Graph Theory, Ph. D. The, Madura Kamara Unverty, Inda. [5] P. Ederg, Decompoton of the Compete Graph wth an Even Number of Vertce, M. Sc. The, McMater Unverty, 997 [6] K. Kathrean, Two cae of gracefu graph, Ar Combn, 55 (2000) [7] C. Sekar, Stude n Graph Theory, Ph. D. The, Madura Kamara Unverty, Fgure π repreentaton of G Fgure 2; the -repreentaton of the graph P nm ; Fgure 3 3;5 P odd gracefu. Fgure 3 P7;6 odd gracefu ISSN: ISBN:

5 Proceedng of the 9th WSEAS Internatona Conference on APPLIED INFORMATICS AND COMMUNICATIONS (AIC '09) Fgure 4 5;5 P odd gracefu Fgure 4 P9;7 odd gracefu (A)The graph 0;5 ha even number of edge odd gracefu (B) The graph 9;5 ha odd number of edge odd gracefu Fgure 5 (A) (B) (A) The graph P 5 P 3 odd gracefu, (B) The track of the abeng proce Fgure 6 ISSN: ISBN:

6 Proceedng of the 9th WSEAS Internatona Conference on APPLIED INFORMATICS AND COMMUNICATIONS (AIC '09) Fgure 7 Fgure 8 p7 p 6 p8 p5 Fgure 9 ISSN: ISBN: