Fibonacci and Super Fibonacci Graceful Labeling of Some Graphs*

Transcription

1 Studie i Mathematical Sciece Vol, No, 0, pp 4-35 wwwccaadaorg ISSN [Prit] ISSN [Olie] wwwccaadaet iboacci ad Super iboacci Graceful Labelig of Some Graph* SKVaidya, * PLVihol Abtract: I the preet work we dicu the exitece ad o-exitece of iboacci ad uper iboacci graceful labelig for certai graph We alo how that the graph obtaied by witchig a vertex i cycle C, (where 6 ) i ot uper iboacci graceful but it ca be embedded a a iduced ubgraph of a uper iboacci graceful graph Key word: Graceful Labelig; iboacci Graceful Labelig; Super iboacci Graceful Labelig INTRODUCTION Graph labelig where the vertice are aiged value ubject to certai coditio The problem ariig from the effort to tudy variou labelig cheme of the elemet of a graph i a potetial area of challege Mot of the labelig techiue foud their origi with 'graceful labelig' itroduced by Roa (967) The famou graceful tree cojecture ad may illutriou work o graceful graph brought a tide of differet graph labelig techiue Some of them are Harmoiou labelig, Elegat labelig, Edge graceful labelig, Odd graceful labelig etc A compreheive urvey o graph labelig i give i Gallia (00) The preet work i aimed to provide iboacci graceful labelig of ome graph Throughout thi work graph G=( V( G), E( G )) we mea a imple, fiite, coected ad udirected graph with p vertice ad edge or tadard termiology ad otatio i graph theory we follow Gro ad Yelle (998) while for umber theory we follow Nive ad Zuckerma (97) We will give brief ummary of defiitio ad other iformatio which are ueful for the preet ivetigatio Defiitio A vertex witchig G v of a graph G i obtaied by takig a vertex v of G, removig all edge icidece to v ad addig edge joiig v to every vertex which are ot adjacet to v i G Departmet of Mathematic, Saurahtra Uiverity, Rajkot , Gujarat (Idia) amirkvaidya@yahoocoi * Correpodig Author Departmet of Mathematic, Govermet Polytechic, Rajkot , Gujarat (Idia) * AMS Subject claificatio umber(00): 05C78 Received December, 00; accepted April 9, 0 4

2 SKVaidya; PLVihol /Studie i Mathematical Sciece Vol No, 0 Defiitio Coider two copie of fa ( = P K ) ad defie a ew graph kow a joit um of i the graph obtaied by coectig a vertex of firt copy with a vertex of ecod copy Defiitio 3 A fuctio f i called graceful labelig of graph if f : VG ( ) {0,,, } ijective ad the iduced fuctio f : EG ( ) {,, } f ( e= uv)= f( u) f( v) i bijective A graph G i called graceful if it admit graceful labelig Defiitio 4 The iboacci umber 0,, = i defied a are defied by 0,, Defiitio 5 The fuctio f : VG ( ) {0,,, } (where i the ad th iboacci umber) i aid to be iboacci graceful if f : EG ( ) {,, } defied by f ( uv)= f ( u) f ( v) i bijective th Defiitio 6 The fuctio f : VG ( ) {0,,, } (where i the iboacci umber) i aid to be Super iboacci graceful if the iduced edge labelig f : EG ( ) {,, } defied by f ( uv)= f ( u) f ( v) i bijective Above two cocept were itroduced by Kathiree ad Amutha [5] Deviatig from the defiitio they aume that,, 3 3, 4 5 ad proved that K i iboacci graceful if ad oly if 3 If G i Euleria ad iboacci graceful the 0( mod3) Every path P of legth i iboacci graceful P i a iboacci graceful graph Caterpillar are iboacci graceful The bitar B m, i iboacci graceful but ot Super iboacci graceful for 5 C i Super iboacci graceful if ad oly if 0( mod3) Every fa i Super iboacci graceful If G i iboacci or Super iboacci graceful the it pedat edge exteio G i iboacci graceful If G ad G are Super iboacci graceful i which o two adjacet vertice have the labelig G G i iboacci graceful ad, the their uio If G, G,, G are uper iboacci graceful graph i which o two adjacet vertice are labeled with ad the amalgamatio of G, G,, G obtaied by idetifyig the vertice havig label 0 i alo a uper iboacci graceful I the preet work we prove that 5

3 SKVaidya; PLVihol /Studie i Mathematical Sciece Vol No, 0 Tree are iboacci graceful Wheel are ot iboacci graceful Helm are ot iboacci graceful The graph obtaied by Switchig of a vertex i a cycle C i iboacci graceful Joit Sum of two copie of fa i iboacci graceful Switchig of a vertex i a cycle C i uper iboacci graceful except 6 Switchig a vertex of cycle C for 6 ca be embeded a a iduced ubgraph of a uper iboacci graceful graph Obervatio 7 If i a triagle edge receive iboacci umber from vertex label tha they are alway coecutive MAIN RESULTS Theorem Tree are iboacci graceful Proof: Coider a vertex with miimum eccetricity a the root of tree T Let thi vertex be v Without lo of geerality at each level of tree T we iitiate the labelig from left to right Let 3 P, P, P, P be the childre of v Defie f : VT ( ) {0,, } i the followig maer f()=0 v, ( )= f P Now if P i t are childre of ( ) i f ( P )= f( P ) i i, i t If there are r vertice at level two of label them a follow, f ( P )= f( P ) i t i ir, P the P ad out of thee r vertice, r be the childre of Let there are r vertice, which are childre of P the label them a follow, f( P )= f( P ) i t r i, i r ollowig the ame procedure to label all the vertice of a ubtree with root a P we ca aig label to each vertex of the ubtree with root a 3 i f ( P )= fi, where f i the i P the P, P, P ad defie th f i iboacci umber aig to the lat edge of the tree rooted at i P Now for the vertex P Defie ( )= f P 6

4 SKVaidya; PLVihol /Studie i Mathematical Sciece Vol No, 0 th Let u deote P ij, where i i the level of vertex ad j i umber of vertice at i level At thi tage oe ha to be cautiou to avoid the repeatatio of vertex label i right mot brach or that we firt aig vertex label to that vertex which i adjacet to ad i a iteral vertex of the path whoe legth i larget amog all the path whoe origi i (That i, i a root) Without lo of geerality we coider thi path to be a left mot path to ad cotiue label aigmet from left to right a tated erlier If ( ) P i be the childre of P the defie i f ( P i )= f( P ) i, i If there are P ( ) i i b vertice at level two of of P The label them a follow f( Pi)= f( P) i, i b P ad out of thee b vertice, b be the childre If there are b vertice, which are childre of P the label them a follow, f( P )= f( P ) ( b i) b i, i b We will alo coider the ituatio whe all the vertice of ubtree rooted at i havig all the vertice th of degree two after i level the we defie labelig a follow f( P )= f( P ) ( ) i i ( i) ( labeled vertice i the brach) Cotiuig i thi fahio ule all the vertice of a ubtree with root a P are labeled Thu we have labeled all the vertice of each level That i, T admit iboacci Graceful Labelig That i, tree are iboacci Graceful The followig igure will provide better uder tadig of the above defied labelig patter igure : A Tree Ad it iboacci Graceful Labelig Theorem Wheel are ot iboacci graceful Proof: Let v be the apex vertex of the wheel W ad v v, v be the rim vertice 7

5 SKVaidya; PLVihol /Studie i Mathematical Sciece Vol No, 0 f : VW ( ) {0,, } Defie We coider followig cae f()=0 v Cae : Let o, the vertice v, v v mut be label with iboacci umber f ( v )= Let the f ( v ) = or f ( v ) = f ( v)= If f ( v )= the i ot poible a )= f( vv)= f ( v)= If f ( v ) the )= f( vv)= otherwie f ( v)= p If be the iboacci umber other the ad the f ( v ) f( v ) = p ca ot be iboacci umber for p > v Cae : If f( v i a rim vertex the defie )=0 f ( v)= If the the apex vertex mut be labeled with or f ()= v Sub Cae : Let Now f ( v ) mut be labeled with either by f ( v)= the )= f( vv)= or by 3 If ad if f ( v)= 3 )= ( the f vv)= f ()= v Sub Cae : Let Now f ( v ) mut be label with either by f ( v)= the )= f( vv)= if f ( v)= 3 if the )= )= )= )= )= 4 3 or by 3 or by 4 8

6 SKVaidya; PLVihol /Studie i Mathematical Sciece Vol No, 0 or W 3, 3) ca ot be iboacci umber Now for >3 let u aume that f v3 f ( v3)= which i ot iboacci umber becaue for )= 3)=, we have ow we have followig cae () < < k k < <, () I () we have k = k = ' = )= f( vv) the 3 I () we have k = k = = i poible oly whe = ad = 3, ad 3)= ) ( )= k = = i poible oly whe = ad = 3, the 3)= f( vv) ad 3)= ) f ( v Thu, we ca ot fid a umber 3)= k for which 3) ad 3) are the ditict iboacci umber or f ( v)= 4 we ca argue a above Sub Cae 3: If f ()= v f ( v The we do ot have two iboacci umber correpodig to ) will receive ditict iboacci umber Thu we coclude that wheel are ot iboacci graceful Theorem 3 Helm are ot iboacci graceful ad ( ) f v uch that the edge v Proof: Let H be the helm ad, v, v 3 v be the pedat vertice correpodig to it If 0 i the label of ay of the rim vertice of wheel correpodig to H the all the poibilitie to admit iboacci graceful labelig i ruled out a we argued i above Theorem Thu poibilitie of 0 beig the label of ay of the pedat vertice i remaied at our dipoal f : V( H) {0,, } Defie f( v Without lo of geerality we aume )=0 the f ( v )= f ( v Let )= p ad f ()= v r 9

7 SKVaidya; PLVihol /Studie i Mathematical Sciece Vol No, 0 I the followig igure () to (3) the poible labelig i demotrated I firt two arragemet the poibility of H 3 beig iboacci graceful i wahed out by the imilar arragemet for wheel are ot iboacci graceful held i Theorem or the remaiig arragemet a how i igure (3) we have to coider followig two poibilitie igure : Ordiary Labelig i H3 Cae : p = r = p < r < r p = the =0 = Cae : p = r = r < p < p r = ' the =0 = Now let f ( v3)= t the coider the cae = = r r rom thee two euatio we have = = r r p < r < t <, o we have r < r < < < ad they are coecutive iboacci umber accordig to Obervatio 7 or r p, t we have = ad = o we have r r 30

8 SKVaidya; PLVihol /Studie i Mathematical Sciece Vol No, 0 = = ad r r which i ot poible imilar argumet ca be made for r p, t ie we have either p < r < t or t < r < p A < <, o we ca ay that with )= the edge of the triagle with vertice f () v, f ( v ) ad f ( v 3) will ot have iboacci umber uch that = um of two iboacci umber t < r < p < Similar argumet ca alo be made for Hece Helm are ot iboacci graceful graph Theorem 4 The graph obtaied by witchig of a vertex i cycle C admit iboacci graceful labelig v, v, v, v Proof: Let 3 C be the vertice of cycle C ad be the graph reulted from v witchig of the vertex f : VC ( ) {0,, } Defie a follow f( v )=0 f( v )= f v ( 3)= f ( vi 3)= i, i 3 Above defied fuctio f admit iboacci graceful labelig Hece we have the reult Illutratio 5 Coider the graph C 8 The iboacci graceful labelig i a how i igure 3 igure 3: iboacci Graceful Labelig of C 8 P K Theorem 6 The graph obtaied by joit um of two copie of fa ( = ) i iboacci graceful 3

9 SKVaidya; PLVihol /Studie i Mathematical Sciece Vol No, 0 v, v, v Proof: Let v ad v, v, 3 v m be the vertice of ad m repectively Let v be the apex vertex of ad v be the apex vertex of m ad let G be the joit um of two fa f : V( G) {0,, } Defie a follow f()=0 v f ( v)= f ( vi)= i, i f ( v )= f( v )= f( v )=,im i i I view of the above defied patter the graph G admit iboacci graceful labelig Illutratio 7 Coider the Joit Sum of two copie of 4 The iboacci graceful labelig i a how i igure 4 igure 4: iboacci Graceful Labelig of Joit Sum of 4 Theorem 8 The graph obtaied by Switchig of a vertex i a cycle except 6 Proof: We coider here two cae =3,4,5 C i uper iboacci graceful cae : or =3 the graph obtaied by witchig of a vertex i a dicoected graph which i ot deirable for the iboacci graceful labelig Super iboacci graceful labelig of witchig of a vertex i C =4,5 for i a how i igure 5 3

10 SKVaidya; PLVihol /Studie i Mathematical Sciece Vol No, 0 igure 5: Switchig of a Vertex i C4 ad C5 ad Super ibobacci Graceful Labelig cae : 6 The graph how i igure 6 will be the ubgraph of all the graph obtaied by witchig of a vertex i C( 6) igure 6: Switchig of a Vertex i C 6 I igure 7 all the poible aigmet of vertex label i how which demotrate the repetitio of edge label igure 7: Poible Label Aigmet for the Graph Obtaied by Vertex Switchig i C 6 () I ig8(a) edge label i repeated a = & 0 = () I ig8(b) edge label i repeated a 33

11 SKVaidya; PLVihol /Studie i Mathematical Sciece Vol No, 0 = & 0 = (3) I ig8(c) edge label p i repeated a p p = p & p 0 = p, where p i ay iboacci umber (4) I ig8(d) edge label p i repeated a p p = p & p 0 = p, where p i ay iboacci umber (5) I ig8(e) edge label p i repeated a p p = p & p 0 =, where p p i ay iboacci umber (6) I ig8(f) edge label i repeated a 3 = & = (7) I ig8(g) edge label i repeated a = & 0 = (8) I ig8(h) edge label i repeated a = & 0 = Theorem 9 The graph obtaied by Switchig of a vertex i cycle C for 6 ca be embedded a a iduced ubgraph of a uper iboacci graceful graph Proof: Let v, v, v3 v be the vertice of C ad v be the witched vertex Defie f : V( G) {0,, 3} f( v )=0 f ( vi )= i, i Now it remai to aig iboacci umber, ad 3 Put 3 vertice i the graph Joi firt vertex v labeled with to the vertex v 3 Now joi ecod vertex v labeled with 3 vertex v ad vertex v labeled with to the vertex v to the Thu the reultat graph i a uper iboacci graceful graph Illutratio 0 I the followig igure 8 the graph obtaied by witchig of a vertex i cycle C 6 ad it uper iboacci graceful labelig of it embeddig i how 34

12 SKVaidya; PLVihol /Studie i Mathematical Sciece Vol No, 0 igure 8: A Super iboacci Graceful Embeddig 3 CONCLUDING REMARKS Here we have cotributed eve ew reult to the theory of iboacci graceful graph It ha bee proved that tree, vertex witchig of cycle C, joit um of two fa are iboacci graceful while wheel ad helm are ot iboacci graceful We have alo dicued uper iboacci graceful labelig ad how that the graph obtaied by witchig of a vertex i cycle ( 6) C doe ot admit uper iboacci graceful labelig but it ca be embedded a a iduced ubgraph of a uper iboacci graceful graph REERENCES [] Gallia, JA,(00) A Dyamic Survey of Graph Labelig The Electroic Joural of Combiatoric,7, # DS 6 [] Gro, J, & Yelle, J (998) Graph Theory ad It Applicatio CRC Pre [3] Kathirea, KM & Amutha,S iboacci Graceful Graph Accepted for Publicatio i Ar Combi [4] Nive,I & Zuckerma,H,(97) A Itroductio to the Theory of Number New Delhi: Wiley Eater [5] Roa, A(967) O Certai Valuatio of the Vertice of a Graph Theory of Graph, (Iterat Sympoium, Rome, July 966) Gorda ad Breach, NY ad Duod Pari