Some Results on Vertex Equitable Labeling

Transcription

1 Ope Joural of Dscrete Mathematcs, 0,, Publshed Ole Aprl 0 ( Some Results o Vertex Equtable Labelg P Jeyath, A Maheswar Research Cetre, Departmet of Mathematcs, Govdammal Adtaar College for Wome, Truchedur, Ida Departmet of Mathematcs, Kamara College of Egeerg ad Techology, Vrudhuagar, Ida Emal: eyaeyath@redffmalcom, bala_th@yahooco Receved December 5, 0; revsed Jauary 3, 0; accepted February 8, 0 ABSTRACT q Let G be a graph wth p vertces ad q edges ad let A 0,,,, A vertex labelg f : V G A s sad to b e a vertex equtable labelg of G f t duces a edge labelg f gve by f uv f u f v such that ad,,3,,, where vf a vf b f E q vf a s the umber of vertces v wth f v a for a A A graph G s sad to be a vertex equtable graph f t admts ve rtex equtable labelg I ths paper, we establsh the vertex equtable labelg of a T p -tree, T K where T s a T p -tree wth eve umber of vertces, bstar B,, the caterpllar Sx, x,, x C K, P ad crow Keywords: Vertex Equtable Labelg; Vertex Equtable Graph Itroducto All graphs cosdered here are smple, fte, coected ad udrected We follow the basc otatos ad termologes of graph theory as [] The symbols V G ad EG deote the vertex set ad the edge se t of a graph G Let Gp, q be a graph wth p V G vertces ad q EG edges A labelg f of a graph G s a mappg that assgs elemets of a graph to the set of umbers (usually to postve or o-egatve tegers) If the doma of the mappg s the set of vertces (the set of edges) the we call the labelg vertex labelg (edge labelg) The labels of the vertces duce labels of the edges There are several types of labelg A detaled survey of graph labelg ca be foud [] A vertex labelg f s sad to be dfferece labelg f t duces the label f x f y for each edge xy whch s called as wegh t of the edge xy A dfferece labelg f of a graph G s sad to be k-equtable f for each weght duced by f o the edges of G appears exactly k tmes If a graph G has a k-equtable labelg the G s sad to be k-equtable Equtable labelg of graphs was troduced by Bloom ad Ruz [3] A bref summary of deftos whch are useful for the preset study s gve below Defto [4] Let T be a tree ad u 0 ad v0 be two adacet vertces T Let u ad v be two pedat vertces of T such that the legth of the path u 0 -u s equal Copyrght 0 ScRes to the legth of the path v0 -v If the edge uv 0 0 s desuch a trasformato of T s called a elemetary par- leted from T ad u ad v are oed by a edge uv, the allel trasformato (or a ept, for short) ad the edge uv 0 0 s called trasformable edge If by a sequece of ept s, T ca be reduce d to a path, the T s called a T p tree (trasformed tree) ad such sequece s regarded as a composto of mappgs (ept s) deoted by P, s called a parallel trasformato of T The path, the mage of T uder P s deoted as P(T) A T p tree ad a sequece of two ept s reducg t to a path are llustrated Fgure Defto The coroa G G of the graphs G ad G s obtaed by takg oe copy of G (wth p vertces) ad p copes of G ad the og the vertex th th of G to every vertex of the copy of G Defto 3 Caterpllar s a tree wth the property that the removal of ts pedat vertces leaves a path Defto 4 The square graph G of a graph G has the vertex set VG VG wth uv, adacet G wheever du, v G x deotes the smallest teger greater tha or equal to x The cocept of mea labelg was troduced by S Somasudaram ad R Pora [5] ad further studed [6-8] A Lourdusamy ad M Seevasa troduced a vertex equtable labelg [9] I a vertex equtable labelg we use the labels 0,,,, q for the vertces,

2 5 P JEYANTHI ET AL Fgure A T p -tree ad a sequece of two ept s reducg t to a path the umber of tmes the dfferet vertex labels appear caot dffer by more tha oe The duced edge labels are defed as the sum of the cdet vertex labels They proved that the graphs lke path, bstar B,, combs P, K bpartte complete K,, fredshp t graph C 3 for t, quadrlateral sake, K mk, K, K, k f ad oly f k 3, ladder graph L P K, arbtrary super dvso of a path ad cycle C wth 0 or 3mod 4 are vertex equtable Also they proved that the graph K, f 4, Eulera graph wth edges where or mod 4, the wheel W, the complete graph K f >3 ad tragular cactus wth q edges where q 0 or 6 or 9 mod are ot vertex equtable Moreover they proved that f G s a graph wth p vertces ad q edges, q s eve ad p < q the G s ot vertex equtable Defto 5 [9] Suppose G s a graph wth p vertq edges Let A 0,,,, q A vertex la- ces ad belg f : V A duces a edge labelg f defed by f uv f u f v for all edges uv For a A, let vf a be the umber of vertces v wth f v a A graph G s vertex equtable f there exsts a vertex labelg f such that for all a ad b A, vf av b ad the duced edge labels are,,3,, q P Jeyath ad A Maheswar proved [0,] that tadpoles, C m C, armed crows, [P m ; C ] ad, PoK m,, the graphs obtaed by duplcatg a arbtrary vertex ad a arbtrary edge of a cycl e C, total graph of P, splttg graph of P ad fuso of two edges of a cycle C are vertex equtable graphs I ths paper, we establsh the vertex equtable labelg o f a T p -tree, T K where T s a T p -tree wth eve umber of vertces, the bstar B,, the caterpllar Sx, x,, x ad the crow C, K P f Ma Results, be ay two vertex equtable graphs wth equtable labelg f ad g respectvely Let u ad v be the vertces of G ad G respectvely such that f u ad gv0 The the graph G G obtaed from G f g ad G by detfyg the vertces u ad v s a vertex equtable graph Proof Clearly G G has q edges ad f g p p vertces Let Theorem Let G p ad G p, q Defe V G u, u : p, V G v, v : p q f g h: V G G 0,,,, hu fu for by p, h v f u ad hv gu for p Clearly, vf a f 0 a vh a q vg a f a Therefore, vh a vh b ad the labels of the edges of the copy of G are,,, ad the labels of the edges of the copy of G are,,, q Hece, G G s a f g vertex equtable graph Theorem Let G p, ad Gp, q be ay two vertex equtable graphs wth equtable labelg f ad g respectvely Let u ad v be the vertces of G ad G respectvely such that f u ad gv 0 The the graph G obtaed by og u ad v by a edge s vertex equtable Proof Clearly G has q edges ad p p vertces Let, :,, : V G u u p V G v v p q h: V G 0,,,, Defe Copyrght 0 ScRes

3 P JEYANTHI ET AL 53 by hw f w, f, f w V G The la wv G h w g w bels of the edges of the copy of G are,,, ad the labels of the edges of the copy of G are 3, 4,,q ad h uv h u h v Hece, G s a vertex equtable graph Theorem 3 Every T p -tree s a vertex equtable graph Proof Let T be a T p -tree wth vertces By the defto of a trasformed tree there exsts a parallel trasfor- mato P of T such that for the path PT we have ) VPT VT, ) EPT ET Ed Ep where E d s the set of edges deleted from T ad E p s the set of edges ewly added through the sequece P P, P,, Pk of the epts P used to arrve the path PT Clearly, E d ad E p have the same umber of edges Now deote the vertces of PT successvely as v, v,, v startg from oe pedat vertex of PT rght up to the other For, defe the labelg f as f v f s odd f s eve The f s a vertex equtable labelg of the path PT Let vv be ay edge of T wth < ad P be the ept that deletes ths edge ad add the edge vtv t where t s the dstace of v from v t ad also the dstace of v from v t Let P be a parallel trasfor- of T that cotas P mato as oe of the costtuet epts Sce vtvt s a edge of the path PT, t follows that t t whch mples t Therefore ad ar e of opposte party The duced label of the edge vv s gve by Now t f vv f vv f v f v t t t, t t t t f v v f v v t f v f v t t, f s eve f s odd t Ther efore, we have f vv f vtvt ad hece f s a vertex equtable labelg of the T p -tree T A example for the vertex equtable labelg of a T p - tree wth vertces s gve Fgure Theorem 4 Let T be a Tp-tee wth eve umber of vertces The the graph T K s a vertex equtable graph for all Proof L et T be a T p -tree of eve order m ad the ver- tex set V T v, v, v3,, v m Let u, u,, u be the pedat vertces oed wth v m by a edge The K, :, m V T v u By the defto of a T p -tree, there exsts a parallel trasformat o P of T such that for the path PT we have ) VPT VT, ) EPT ETEd Ep where E d s the set of e dges deleted from T ad Ep s the set of edges ewly added through the sequece P P,,, P Pk of the epts P used to ar rve the path PT Clearly, E d ad E p have the same umber of edges Now deote the vertces of PT successvely as v, v,, vm startg from oe pedat vertex of PT rght up to the other The labelg f defed by f v f f s odd s eve, Fgure Vertex equtable labelg of a T p -tree wth vertces Copyrght 0 ScRes

4 54 P JEYANTHI ET AL f s odd f u, f s eve s a vertex equtable labelg graph Let vv be ay edge of T wth < m let P be th e ept that deletes ths edge ad adds the e vtvt where t s the dstace of v from v ad t dstace of v from v t Let P be a parallel trasformato of T that cotas P as oe of the co- sttuet epts Sce vtv t s a edge the path PT, t follows that t t whch mples t Therefore ad are of opposte party The duced label of the edge vv s gve by Therefore, we have f vv f v s a vertex equtable labelg of T K v ad thus f t t A example for the vertex equtable labelg of T K, 5 where T s a Tp-tree wth vertces s show Fgure 3 Let B, be a graph obtaed from K by attachg pedat edges at oe vertex ad pedat edges at the other vertex Theorem 5 The bstar B, s a vertex equtable graph Proof Let V K u, v ad u ad v be the vertces adacet to u ad v re- spectvely Now, B, has edges ad 3 vertces Defe dge also the t t f vv f vv f v f v t t t, f v v f v v f v f v t t t t t t t, f s eve f s odd f : V B, 0,,,, by f u 0, f v, f u f ad f v f The f s a vertex e qutable labelg of B, Theorem 6 Let x < x x3 x ad x x4 xxx3 x f s eve k xx3 xx x4 x f s odd The S x, x,, x, k graph s a vertex equtable Fgure 3 Vertex equtable labelg of T K 5 Copyrght 0 ScRes

5 P JEYANTHI ET AL 55 Proof By Theorem 5,, S x x s a vertex equtable graph Let f be the correspodg vertex equtable labelg of Sx, x Let y x x Sce x < x, y 0 Cosder the graphs Sx, x ad Sy, y The umber of edges of the graph S x, x s x Now, Sx, x Sy, y f f S x, x y, y S x, x, y Therefore, by Theorem, S x, x y, y s a vertex equtable graph Let f be the correspodg vertex equtable labelg of S x, x, y Aga the umber of edges of Sx, x, y s eve Now take y x3 y x3 x x Hece y 0 Also Sx, x, y Sy, y S x, x, y y, y S x, x, x, y 3 f f Therefore, by Theorem,,,, S x x x3 y s a vertex equtable graph ad the umber of edges s eve Proceedg lke ths, at the th step we get Sx, x, x3,, x, y s a vertex equtable graph where y x x x x x x 4 3 f s eve xx3 x x x4 x f s odd Let f be the correspodg vertex equtable label-, x, x,, x, y Take g of S x 3 x x x x x x y 4 3 f s eve xx3 xx x4 x f s odd Clearly y y x Now, S x, x, x,, x, y S y, y 3 Sx x x x y y y Sx, x, x3,, x, k Therefore, S x, x, x,, x, x, k,,,,,, 3 f f 3 s a vertex equtable graph A example for the vertex equtable labelg of S 4,6,9,7 f s odd s gve Fgure 4 A example for the vertex equtable labelg of S 5,7,9,0, f s eve s gve Fgure 5 Fgure 4 Vertex equtable labelg of S (4, 6, 9, 7 + ) Fgure 5 Vertex equtable labelg of S (5, 7, 9, 0, + ) Copyrght 0 ScRes

6 56 P JEYANTHI ET AL s a vertex equta- Theorem 7 The crow ble graph Proof: Let u, u, u be the vertces of the cycle C ad let v be the vertex adacet to u for The the vertex set C K V C K u, v : Kuu, uv, uu u v : 0,,, ad the edge set E C,: Defe f V C K for the followg cases: Case 0 mod 4 for,3,5,, f u for,4,6,, for, for,3,5,, f v for, 4,6,, for Case mod 4 for,3,5,, f u for, 4,6,, for, for,3,5,, f v for,4,6,, for Case 3 mod 4 for,3,5,, f u for,4,6,, for, Fgure 6 Vertex equtable labelg of C8 K for,3,5,, f v for, 4,6,, for 3 f v f v,, f u 3 mod 4 Case 4 for,3,5,, f u for,4,6,, for, for,3,5,, f v for, 4,6,, for I all the above cases, f s a vertex equtable labelg Hece C K s a vertex equtable graph A example for the vertex equtable labelg of C8 K s show Fgure 6 Theorem 8 The graph P s a vertex equtable graph Proof Let u, u,, u be the path P Clearly, P has vertces ad 3 edges Defe 3 f : VP 0,,,, Copyrght 0 ScRes

7 P JEYANTHI ET AL 57 by f u, Evdetly, equtable graph REFERENCES s a vertex [] F Harary, Graph Theory, Addso Wesley, Massachusetts, 97 [] J A Galla, A Dyamc Survey of Graph Labelg, The Electroc Joural of Combatorcs, Vol 8, 0, Paper #DS6 [3] G Bloom ad S Ruz, Decomposto to Lear Forest ad Dfferece Labelgs of Graphs, Dscrete Appled Mathematcs, Vol 49, 994, pp 6-75 do:006/066-8x(94)900- P [4] S M Hegde ad S Shetty, O Graceful Trees, Appled Mathematcs E-Notes, Vol, 00, pp 9-97 [5] R Pora ad S Somasudram, Mea Labelg of Graphs, Natoal Academy Scece Letters, Vol 6, 003, pp 0-3 [6] R Pora ad S Somasudram, No-Exstece of Mea Labelg for a Wheel, Bullet of Pure ad Appled Sc- atcs & Statstcs), Vol E, 003, pp eces (Mathem 03- [7] R Pora ad S Somasudram, Some Results o Mea Graphs, Pure ad Appled Mathematcal Sceces, Vol 9, 004, pp [8] R Pora ad S Somasudram, Further Results o Mea Graphs, Proceedgs f SACOEFERENCE, Natoal Level Coferece, Dr Svath Adtaar College of Egeerg, 005, pp [9] M Seevasa ad A Lourdusamy, Vertex Equtable Labelg of Graphs, Joural of Dscrete Mathematcal Sceces & Cryptography, Vol, No 6, 008, pp [0] P Jeyath ad A Maheswar, O Vertex equtable labelg, Preprt [] P Jeyath ad A Maheswar, Vertex Equtable Labelg of Cycle ad Path Related Graphs, Utltas Mathematca, Artcle Press Copyrght 0 ScRes