ODD HARMONIOUS LABELINGS OF CYCLIC SNAKES

Transcription

1 ODD HARMONIOUS LABELINGS OF CYCLIC SNAKES ABSTRACT In [8] Lang and Ba hae shon that the M. E. Abdel-Aal Departent of Matheatcs, Faculty of Scence, Benha Unersty, Benha 58, Egypt In ths paper e generalze ths result on cycles by shong that the C snae graph s an odd haronous graph for each. Cn snae th strng,,, hen C snae th -pendant edges for -,. Fnally e present soe exaples to llustrate the proposed n 0 (od ) are odd haronous graph. Also e sho that the each,, (for lnear case and for general case). Moreoer, e sho that, all subdson of snae are odd haronous for each theores. KEYWORDS Odd haronous labelng, pendant edges, Cyclc snaes, Subdson double trangular snaes.. INTRODUCTION Graph labelng s an acte area of research n graph theory hch has anly eoled through ts any applcatons n codng theory, councaton netors, oble telecouncaton syste. Optal crcuts layouts or graph decopostons probles, no nae ust a fe of the. Most graph labelng ethods trace ther orgon to one ntroduced by Rosa [] called such a labelng a β-aluaton and Golob [] subsequently called graceful labelng, and one ntroduced by Graha and Sloane [] called haronous labelng. Seeral nfnte fales of graceful and haronous graphs hae been reported. Many llustrous ors on graceful graphs brought a tde to dfferent ays of labelng the eleents of graph such as odd graceful. A graph G of sze q s odd-graceful, f there s an necton f fro V (G) to {0,,,, q -} such that, hen each edge xy s assgned the label or eght f(x) - f(y), the resultng edge labels are {,, 5,..., q - }. Ths defnton as ntroduced by Gnanaoth []. Many researchers hae studed odd graceful labelng. Seoud and Abdel-Aal [5] they deterne all connected odd graceful graphs of order 6. For a dynac surey of arous graph labelng probles e refer to Gallan [6]. DOI : 0.5/graphoc.0.50

2 Throughout ths or graph G = (V(G), E(G)) e ean a sple, fnte, connected and undrected graph th p ertces and q edges. For all other standard ternology and notons e follo Harary [7]. A graph G s sad to be odd haronous f there exsts an necton f: V(G) {0,,,, q-} such that the nduced functon f * : E(G) {,,..., q } defned by f * (u) = f(u) + f() s a becton. Then f s sad to be an odd haronous labelng of G [8]. A graph Cn snae as ntroduced by Barrentos [9], as generalzaton of the concept of trangular snae ntroduced by Rosa [0]. Let G be a Cn snae th. Let u, u,..., u be the consecute cut-ertces of G. Let d be the dstance beteen u and u+ n G, and the strng ( d, d,..., d ) of ntegers. Hence, any graph G = C snae, can be represented by a strng. For nstance, the n strng (fro left to rght) of the 8C snae on Fgure () s,,,,,. Gracefulness of the nd of C snae studed by Gnanaoth hae strng,,,. And the labelngs gen by Ruz consdered by C snae th strng,,,. We obtan n Theore. an odd haronous labelngs of the C snae th strng,,,. If the strng of gen C snae s n, n,, n, e say that C n snae s lnear. Ths paper can be dded nto to sectons. Secton, e sho that the graphs C snae ʘ (the graph obtaned by onng -pendant edges to each ertex of C snae, ) for lnear and general cases of C snae for each are odd haronous. We also obtan an odd haronous labelng of Cn snae th the sequence strng s,,..., and hen n 0 (od ). In secton, e sho that, an odd haronous labelng of the all subdson of double trangular snaes ( -snae). Fnally, e proe that the all subdson of - snae are odd haronous for each,. n. MAIN RESULTS In [8, Corollary. ] Lang and Ba n Corollary. () hen =, they hae shon that, the C snae graph s an odd haronous graph for each. We extended ths result to obtan an odd haronous labelng for the corona C snae graph (the graph obtaned by onng pendant edges to each ertex C snae,) are denoted by C snae ʘ. Theore.. The lnear graphs C snae ʘ are odd haronous for,

3 Proof. Consder the lnear graph C snae, hch has the ertces, u, and here = 0,,,...,, =,,...,. In order to get the lnear C snae ʘ,,, e l l l add -pendant edges, u, and to each ertex of, u, and respectely such that l = 0,,,...,. No, let G be the lnear C snae ʘ,, be descrbed as ndcated n Fgure u u u u u u u u u u u u Fgure It s clear that the nuber of edges of the graph H s q = + (+). We defne the labelng functon f : V(G) {0,,,, 8 + (+) - } as follos: f ( ) =, 0. f ( ) =,, f ( u ) =,, f ( ) = [ q ( + ) ] +, 0, f ( u ) = ( + ) ( + ) + +,, f ( ) = ( + ) ( + )( ),,. The edge labels ll be as follos: The ertces,,, nduce the edge labels f ( ) + f ( ) = 8 7,. The ertces, u,, nduce the edge labels f ( ) + f ( u ) = 8 5,. The ertces,, nduce the edge labels f ( ) + f ( ) = 8,. The ertces u,, nduce the edge labels f ( u ) + f ( ) = 8,. The reanng odd edge labels fro 8+ to +(+) are obtaned fro the follong f ( u ) + f ( u ) = ( + ) + +,,. f ( ) + f ( ) = ( + ) + +,,,

4 f ( ) + f ( ) = [ q ] +, 0,, So { f ( u) + f ( ) : u E( G) }= {,,5,, q-}. Hence the graph G s odd haronous. Exaple.. An odd haronous labelng of the graph lnear Fgure (). C snae ʘ, s shon n Fgure (): The graph lnear C snae ʘ th ts odd haronous labelng. For the general for of cyclc graph e obtan the follong result. Theore.. The follong graphs are odd haronous () C snae for each, () C snae ʘ for each,, (the general for). Proof. The graph C snae can be consdered as a bpartte graph (one partte set has blac ertces and the other has hte ertces) t s possble to ebed t, on a square grd as s shoed n the next Fgure. Fgure Let G be the graph C snae ʘ hch obtaned by onng - pendant edges to each ertex of C snae. Then V ( G) = ( + )( + ) and E ( G) = + ( + ). No, e are runnng the follong steps sequentally n order to proe the Theore: C Step. Snce snae s a bpartte graph so t has one partte set has blac ertces and the other has hte ertces as s shoed n Fgure (). Put blac ertces n a strng, ordered by dagonals fro left to rght and nsde each dagonal fro botto to top, assgn to the fro an arthetc progresson of dfference hch frst ter s zero, hen e oe to another dagonal

5 e use an arthetc progresson of dfference, countng untl the last blac ertex has been nubered. Slarly, Put the hte ertces on a strng, ordered for dagonals fro left to rght and nsde each dagonal fro top to botto, startng th the frst dagonal assgn nubers fro an arthetc progresson of dfference, hch frst ter s, hen e oe to another dagonal e use an arthetc progresson of dfference, countng untl the last hte ertex has been nubered. Step. In ths step, e are labelng the ertces of -pendant edges hch contact th the hte dagonals, fro rght to left and nsde each hte dagonal fro botto to top, assgn to the fro an arthetc progresson of dfference, hch frst ter s z such that z = y + here y s the last ertex labelng of blac dagonal, hen e oe to a ne ertex of the hte dagonal, the frst ertex of - pendant edges s labeled by an arthetc progresson of dfference, but the arthetc progresson of dfference has been used th the rean ( -)ertces of -pendant edges. We oe fro a ertex to another of the hte dagonals untl the last hte ertex. Step. Fnally, e are labelng the ertces of -pendant edges hch contact th the blac dagonals, fro left to rght and nsde each blac dagonal fro botto to top, assgn to the fro an arthetc progresson of dfference (-), hch frst ter s (q-) here q s the sze of G, hen to oe to a ne ertex of the blac dagonal, the frst ertex of - pendant edges s labeled by an arthetc progresson of dfference ( -6), but the arthetc progresson of dfference(-) has been used th the rean (-)ertces of -pendant edges. We oe fro a ertex to another of the blac dagonals untl the last blac ertex. No, e hae coplete the proof by runnng the aboe steps,.e. e enton only the ertces labels and the reader can fulfll the proof as e dd n the preous theore here step ge us an odd haronous labelng of the graph C snae for each, and step- step ge us an odd haronous labelng of the graph C snae ʘ for each, (n general case). The follong exaple llustrates the last result. Exaple.. An odd haronous labelng of the graph shon n Fgure. 5C snae ʘ (for general case) s Fgure : The graph 5C snae ʘ (for general case) th ts odd haronous labelng. 5

6 The graphs Cn snae hen the sequence strng s (,,,, ) hen n 0 (od ) are studed n the follong Theore: Theore.. The graphs C snae for each,, th strng (,,, ) are odd haronous. Proof. Let G = C snae = ( ) C snae can be descrbed as ndcated n Fgure 5 n ( n ) ( n ) n ( n ) ( n ) n n n ( n ) ( n ) n ( n ) ( n ) n Fgure 5 It s clear that E ( G) = ( n ). We defne the labelng functon f : V(G)) {0,,,, 8(n )-} as follos: ( ), =,,5,... n or n f ( ) = ( ) +, =,,6,... n or n. f ( ) =, cases: Case() For labelng the ertces, n, e consder the follong to f s odd, n e hae the follong labelng, for each f ( ) = f ( ) + Case() f s een, n e hae the follong labelng, for each : f ( ) +, =,,5,... f ( ) = f ( ), =,,6,.... No e label the reanng ertces, n, as follos: f ( ) = f ( ) + ( ),, n. 6

7 It follos that f adts an odd haronous labelng for ( n ) C snae. Hence ( n ) C snae for each n, th the strng (,,,) are odd haronous graphs. Exaple.5. Odd haronous labelng of graph n Fgure 6. C8 snae th the strng (,,,) s shon Fgure 6: The graph C8 snae th ts odd haronous labelng.. SUBDIVISION OF DOUBLE TRIANGLES SNAKE Rosa [0] defned a trangular snae (or -snae) as a connected graph n hch all blocs are trangles and the bloc-cut-pont graph s a path. Let -snae be a -snae th blocs hle n -snae be a -snae th blocs and eery bloc has n nuber of trangles th one coon edge. Dad Moulton [] proed that trangular snaes th p trangles are graceful f p s congruent to 0 or odulo. Xu [] proed that they are haronous f and only f p s congruent to 0, or odulo. A double trangular snae s a graph that fored by to trangular snaes hae a coon path. The haronous labelng of double trangle snae ntroduced by X Yue et al []. It s non that, the graphs hch contan odd cycles are not odd haronous so e used the subdson notaton for odd cycle n order to proe that all subdson of double trangular snaes are odd haronous. Theore.. haronous. All subdson of double trangular snaes ( -snae ) are odd Proof. Let G = -snae has q edges and p ertces. The graph G conssts of the ertces ( u, u,,u + ), (,,, ), (,,, ) therefore e get the subdson of double trangular snaes S(G) by subddng eery edge of double trangular snaes -snae exactly once. Let y be the nely added ertex beteen u and u + hle and are nely added ertces beteen u and u + respectely, here. Fnally, and are nely added ertces beteen 7

8 u and u + respectley, such that. Let the graph S(G) be descrbed as ndcated n Fgure 7 u y u y u y u u y u + Fgure 7: the subdson of double trangular snaes ( -snae). It s clear that the nuber of edges of the graph S(G) s 0. We defne the labelng functon: f : V(S(G)) {0,,,,, 0 - } as follos: f( u ) = 6( - ), + = n, f( y ) = -, = n-, f( ) = 6 +, = n-, f( ) = 6 -, = n-, f( ) = + 8 -, = n-, =,, f( ) = + 6-5, = n-, =,. The edge labels ll be as follos: The ertces u and,, nduce the edge labels {0-9, } = {,,,0-9}. The ertces u and y,, nduce the edge labels{0-7, } = {,,,0-7}. The ertces u and,, nduce the edge labels{0-5, } = {5,5,,0-5}. The ertces and,, nduce the edge labels{0-, } = {7,7,,0-}. The ertces y and u +,, nduce the edge labels{0-, } = {9,9,,0-}. The ertces and,, nduce the edge labels{0-9, } = {,,,0-9}. The ertces and,, nduce the edge labels{0-7, } = {,,,0-7}. The ertces and u +,, nduce the edge labels{0-5, } = {5,5,,0-5}. The ertces and u +,, nduce the edge labels{0-, } = {7,7,,0-}. The ertces and,, nduce the edge labels{0-, } = {9,9,,0-}. So e obtan all the edge labels{,,5,,0-}. Hence the subdson of double trangular snaes ( -snae, ) are odd haronous. Theore.. All subdson of -snae,, are odd haronous. 8

9 Proof. Let G = -snae has q edges and p ertces. The graph G conssts of the ertces ( u, u,,u + ), (,,..., ), (,,..., ),. Therefore e get generalzed the subdson of double trangular snaes S(G) by subddng eery edge of -snae exactly once. Let y be the nely added ertex beteen u and u+ hle and are nely added ertces beteen u and u+ respectely. Fnally, and are nely added ertces beteen u and u + respectely here =,,..., and =,,,..., ( Fgure 8 ). It s Clear that, the nuber of edges of the graph S(G) s q = (8 + ) edges. n ( n ) ( n ) n ( n ) ( n ) n u u u u n ( n ) ( n ) u n y y y ( n ) n ( n ) n ( n ) ( n ) ( n ) n ( n ) n Fgure 8: the subdson of -snae We defne the labelng functon f : V(S(G)) {0,,,,, (8+) - } as follos: <, f( u ) = ( + )( - ), + = n, f( y ) = ( + ) - 0 -, < <, f( ) = ( + )( - ) + l -, < <, < l <, l f( ) =( + 6) + ( + )( - ) +( - ), < <, < <, f( ) = (-)+(+)( - ) + ( + )(l- ) - ( - ), < <, < l <, < <, l f( ) = ( + ) + ( + )( - ) + (8 + )(l - ) - ( - ), l f( ) =( + ) + ( + )( - ) + (6 + )(l - ), < <, < l<. l < <, < l <, < 9

10 In a e of the aboe defned labelng pattern f s odd haronous for the graph S(G). Hence S( -snae) s odd-graceful for all >, >. Illustraton.. An odd haronous labelng of the graph subdson of 6 -snae s shon n Fgure Fgure 9: the graph subdson of 6 -snae th ts odd haronous labelng.. CONCLUSION Haronous and odd haronous of a graph are to entrely dfferent concepts. A graph ay posses one or both of these or nether. In the present or e nestgate seeral fales of odd haronous cyclc snaes. To nestgate slar results for other graph fales and n the context of dfferent labelng technques s open area of research. 5. REFERENCES [] A. Rosa, (967) On certan aluatons of the ertces of a graph, n Theory of Graphs, Internatonal Syposu, Roe, July 966, Gordon and Breach, NeYor and Dunod, Pars, pp [] S.W. Golob, (97) Ho to nuber a graph, n Graph Theory and Coputng, R.C. Read, ed.,acadec Press, NeYor, pp. 7. [] R. L. Graha, N. J. A. Sloane, (980) On addte bases and haronous graphs, SIAM J. Algebr. Dsc. Math., Vol, No, pp [] R.B. Gnanaoth, (99) Topcs n graph theory, Ph.D. thess, Madura Kaara Unersty, Inda. [5] M.A. Seoud and M.E. Abdel-Aal, (0) On odd graceful graphs, Ars Cobn., 08, pp [6] J. A. Gallan. ( 0) A Dynac Surey of Graph Labelng, Electronc J. Cobn. Fourteenth edton. [7] F. Harary. (969) GpaphTheory, Addson-Wesley, Readng MA. 0

11 [8] Z. Lang, Z. Ba. (009) On the odd haronous graphs th applcatons, J. Appl.Math. Coput., 9, pp [9] C. Barrentos, (00) Dfference Vertex Labelngs, Ph.D.Thess, Unersty Poltecnca De Cataunya, Span. [0] A. Rosa, (967) Cyclc Stener Trple Syste and Labelngs of Trangular Cact, Scenta, 5, pp [] D. Moulton, (989) Graceful labelngs of trangular snaes, Ars Cobn., 8, pp. -. [] S.D. Xu, (995) Haroncty of trangular snaes, J. Math. Res. Exposton, 5, pp [] X Yue, Yang Yuansheng and Wang Lpng, (008 ) On haronous labelng of the double trangular n snae, Indan J. pure apple. Math.,9(), pp Author Mohaed Elsayed Abdel-Aal receed the B.Sc. (Matheatcs) the M.Sc.(Pure Matheatcs-Abstract Algebra) degree fro Benha Unersty, Benha, Egypt n 999, 005 respectely. Also, he receed Ph.D. (Pure Matheatcs) degree fro Faculty of Matheatcs, Ta Natonal Unersty, Tastan, n 0. He s a Unersty lecturer of Pure Matheatcs th the Benha Unersty, Faculty of Scence, Departent of Pure Matheatcs. Hs current research s Ordnary partal dfferental equatons, Graph Theory and Abstract Algebra.