EDGE- ODD Gracefulness of the Tripartite Graph

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1 EDGE- ODD Graceuless o the Trpartte Graph C. Vmala, A. Saskala, K. Ruba 3, Asso. Pro, Departmet o Mathematcs, Peryar Maamma Uversty, Vallam, Thajavur Post.. Taml Nadu, Ida. 3 M. Phl Scholar, Departmet o Mathematcs, Peryar Maamma Uversty, Vallam, Thajavur Post.. Taml Nadu, Ida. ABSTRACT A p, q) coected graph s edge-odd graceul graph there exsts a jectve map : EG) {, 3,, q-} so that duced map + : VG) {0,,, 3,, k-)} deed by x ) { x, y ) / xy E } mod k),where the vertex x s cdet wth other vertex y ad k = max {p,q} makes all the edges dstct ad odd. I ths artcle, the Edge odd graceuless o the trpartte graph K, K, K K,,, 3,,, or odd ad K, 4,, 5, or eve are obtaed. Key words: Graceul Graph, Edge-odd graceul labelg, Edge-odd Graceul Graph. INTRODUCTION AND BASIC DEFINITIONS Graph labelg s a assgmet o tegers to the vertces or edges, or both, subject to certa codtos. Graph labelgs were rst troduced the late 90s. I the terveg years dozes o graph labelgs techques have bee studed over 800 papers. Graph labelg have ote bee motvated by practcal problems, s oe o ascatg areas o research. A systematc study o varous applcatos o graph labelg s carred out Bloom ad Golomb[]. Labeled graph plays vtal role to determe optmal crcut layouts or computers ad or the represetato o compressed data structure. The study o graceul graphs ad graceul labelg methods was troduced by Rosa []. Rosa deed a valuato o a graph G wth q edges as a jecto rom the vertces o G to the set {0,,,, q} such that whe each edge xy s assged the label x)-y), the resultg edge labels are dstct. Valuatos are the uctos that produce graceul labelg. However, the term graceul labelg was ot used utl Golomb studed such labelg several years later [3]. For a dyamc survey o varous graph labelg problems alog wth a extesve bblography we reer to Galla [4]. We beg wth smple, te, coected ad udrected graph G = V, E) wth p vertces ad q edges. For all other stadard termology ad otos we ollow Harary[5]. Deto : Graph A graph s a ordered trple G V G ), E G ), I ), where V G ) s a oempty set, E G ) s a set G dsjot rom V G ), ad I s a cdece map that assocates wth each elemet o E G ) G par o elemets o V G )., a uordered Deto : Smple Graph A graph s sad to be smple t has o loops ad o multple or parallel edges. 0

2 Deto 3: Complete Graph A smple graph G s sad to be complete every par o dstct vertces o G are adjacet G. Ay two complete graphs each o vertces are somorphc; each such graph s deoted by Deto 4: The complete K-partte graph A complete k-partte graph s a k-partte graph whch there s a edge betwee every par o vertces rom deret depedet sets. These graphs are descrbed by otato wth a captal letter K subscrpted by a sequece o the szes o each set the partto. For stace, K,, s the complete trpartte graph o a regular octahedro, whch ca be parttoed to three depedet sets each cosstg o two opposte vertces. Deto 5: The complete trpartte graph A complete trpartte graph s the k = 3 case o a complete k-partte graph. I other words, t s a trpartte graph.e., a set o graph vertces decomposed to three dsjot sets such that o two graph vertces wth the same set are adjacet) such that every vertex o each set graph vertces s adjacet to every vertex the other two sets. I there are,, ad graph vertces the three sets, the complete trpartte graph sometmes also called a complete trgraph) s deoted by k p,q,r. Deto : Graceul Graph A ucto o a graph G s called a graceul labelg wth m edges, s a jecto rom the vertex set o G to the set {0,,,, m} such that whe each edge uv s assged the label u) v) ad the resultg edge labels are dstct. The the graph G s graceul Deto 7: Edge Graceul Graph K. A graph GV, E) s sad to be edge-graceul there exsts a bjecto rom E to {,,., E } such that the duced mappg + rom V to {0,,., V -} gve by x ) xy ) mod V take over all edges xy s a bjecto. Deto 8: Odd Graceul Graph A graph G wth q edges to be odd graceul there s a jecto rom V G) to {0,,,, q-} such that, whe each edge xy s assged the label x)-y), the resultg edge labels are {, 3, 5,, q -}. Deto 9: Edge-odd graceul graph A p, q) coected graph s edge-odd graceul graph there exsts a jectve map : EG) {, 3,, q-} so that duced map + : VG) {0,,,,k-)} deed by + x) {x, y)/xy }mod k), where the vertex x s cdet wth other vertex y ad k = max {p, q} makes all the edges dstct ad odd. Hece the graph G s edge - odd graceul.. MAIN RESULT I ths secto, the edge - odd graceuless o complete the trpartte graphs K, K, K, K,,, 3,,, or odd ad K, 4,, 5, Theorem : or eve are obtaed. The Complete trpartte graph K,, s edge-odd graceul. Proo: The graph K,, s a complete trpartte graph wth +4 vertces ad 4+4 edges ad the arbtrary labelgs o K,, are metoed below

3 Fgure : Edge-Odd graceul graph K,, Case : = 4, 8,,, To d edge odd graceul, dee : E K ) {, 3, 5,, q-} or beg multples o 4 as ollows:,, e ),, 4, 5,..., 4 4. e ) 5, e ) 3 Case : 4, 8,,, Fgure : Edge Odd graceul graph K,, 4 To d edge odd graceul, dee : E K ) {, 3, 5,, q-} or 4, 8,,, as ollows:,, e ),,, 3,..., 4 4. The above deed edge labelg ucto wll duce the bjectve vertex labelg ucto : V G ) { 0,,,..., k } such that x ) { x, y ) / xy E } mod k), where k = max{ p, q }. Hece K,, admts edge - odd graceul labelg. Theorem : The complete trpartte graph K, 3, s edge odd graceul. Proo: The graph K, 3, s a complete trpartte graph wth +5 vertces ad 5+ edges ad the arbtrary labelg o K, 3, are metoed below.

4 Fgure 3: Edge Odd graceul graph K, 3, To d edge odd graceul, dee E K ) {, 3, 5,..., q } as ollows :, 3, case ) 0 mod ) e ),,, 3,..., 5 case ) mod ) e ), 3, 4,....,, 8,..., 3, 3 8,..., 5 e ) 5, e ) 7 e ) 3, e ) case 3) mod ) case 5) 4 mod ) e ), 3, 4, 5, 7, 8,..., 5 5 e ),,, 3,..., 5 case 4) 3 mod ) e ),,, 3,..., 5 e ) 3, e ) e ) 0, e ) 5 case ) 5 mod ) e ),, 4, 5,, 8, 0,..., 5 9 e ) 7, e ) 7 3 e ) 5, e ) 3 The above deed edge labelg ucto wll duce the bjectve vertex labelg ucto : V G ) { 0,,,...., k } such that x ) { x, y ) / xy E } mod k), where k = max{ p, q }. Hece K, 3,, admts edge - odd graceul labelg Theorem 3: The complete trpartte graph K,, s edge odd graceul. Proo: The graph K,, s a complete trpartte graph wth +8 vertces ad 8+ edges ad the arbtrary labelgs o K are metoed below.,, 3

5 Fgure 4: Edge Odd graceul graph K,, To d edge odd graceul, dee E K ) {, 3, 5,..., q } as ollows. :,, Case ) : 0 mod 4) e ),,, 3,..., 8 Case ) : mod 4) e ) 3, 8 ) 3 e ) 4 5, e ) 3 e ),, 3,...,, 3,...,, 4,..., 8 Case 3) : mod 4) 3 e ) 4 5, e ) e ) 4 5, e ) 3 7 e ),, 3,...,, 8,...,, 4,..., 8 Case 4) : 3 mod 4) e ) 3, e ) 8 e ),, 3, 4,..., 8 The above deed edge labelg ucto wll duce the bjectve vertex labelg ucto : V G ) { 0,,,..., k } such that x ) { x, y ) / xy E mod k), where k = max{ p, q }. Hece K,, } admts edge - odd graceul labelg. Theorem 4 The complete trpartte graph K, 5, s edge odd graceul or eve. Proo The graph K, 5, s a complete trpartte graph wth +7 vertces ad 7+0 edges ad the arbtrary labelgs o the graph s metoed below. 4

6 Fgure 5 Edge Odd graceul graph K, 5, To d edge odd graceul, dee E K ) {, 3, 5,..., q }, or eve as ollows. :, 5, 5 e ) 9 e ) 4 e ) 7 e ) 3 e 0 ) 9 e ) 0 e ) 4 9 e ) e ) e ) 9 The above deed edge labelg ucto wll duce the bjectve vertex labelg ucto : V G ) { 0,,,..., k } such that x ) { x, y ) / xy E mod k), where k = max{ p, q }. Hece K, 5, Theorem 5 } or eve admts edge - odd graceul labelg. The complete trpartte graph K,4, s edge odd graceul or odd. Proo The complete trpartte graph K, 4, s a coected graph wth + vertces ad +8 edges ad the arbtrary labelgs o the graph are metoed below. Fgure Edge Odd graceul graph K, 4, 5

7 To d edge odd graceul, dee E K ) {, 3, 5,..., q }, or odd as ollows. :, 4, e ),, 3, 4,, 7, 8, 0,...., 8 5 e ) 9 e ) e ) 7 e ) 9 9 The above deed edge labelg ucto wll duce the bjectve vertex labelg ucto : V G ) { 0,,,..., k } such that x ) { x, y ) / xy E } mod k), where k = max{p, q}. Hece K,4,, or odd admts edge - odd graceul labelg. REFERENCES []. Rosa A, O certa valuatos o the vertces o a graph, Iteratoal symposum, Rome, July 9, Gordo ad Breach, N.Y ad Duod Pars, pp ,97. []. G.S. Bloom ad S.W. Golomb 977) Applcatos o umbered udrected graphs, Proc. IEEE. Vol 5, pp [3]. S.W. Golomb 97), How to umber a graph Graph theory ad computg, R. C. ed. Academc Press, New York, pp 3-7. [4]. Joeph A.Galla, A dyamc survey o Graph Labeg, The Electroc Joural o combatorcs 05), #D5, Eghteeth edto, Dec [5]. F. Harary, 99), Graph Theory, Addso Weslely. []. A.Solaraju ad K.Chtra, Edge-odd graceul labelg o some graphs, Electrocs Notes Dscrete Mathematcs, Volume 33, Aprl 009, Pages 5 0. [7]. A.Solaraju, A.Saskala, C.Vmala, Edge-odd Graceuless o Cartesa product o C 3 ad C, Iteratoal Joural o Computer Applcatos Volume 0, November 00, Pages 7-9. [8]. A.Solaraju, A.Saskala, C.Vmala, Edge-odd Graceuless o C 3 ʘ P ad C 3 ʘP, Iteratoal Joural o Computer Applcatos Volume 0, November 00, Pages -3.