On Complementary Edge Magic Labeling of Certain Graphs

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1 Amerca Joural of Mathematcs ad Statstcs 0, (3: -6 DOI: 0.593/j.ajms O Complemetary Edge Magc Labelg of Certa Graphs Sajay Roy,*, D.G. Akka Research Scholar, Ida Academc Dea, HMS sttute of Techology, Tumkur, Karataka, Ida Abstract By G(p, q we deote a graph havg p vertces ad q edges, by V(G ad E(G the vertex set ad the edge set of G respectvely. But the vertces ad edges are called the elemets of the graph. A (p, q graph G s called the edge magc f there exsts a bjectve fucto f: V(G U E(G {,,,p+q} such that f(u+f(v+f(uv=k s a costat called the valece of f for ay edge uv of G. Gve a edge magc f of a graph G(p, q the fucto f( x such that f( x=p+q+-f(x for all elemets of G s sad to be complemetary to f(x or complemetary edge magc labelg f( x. The purpose of ths artcle s to search for certa graphs Km, (m,, C ( 3, p, f (fa B (bwk ad G ( where G s bpartte or trpartte whch have complemetary edge magc stregth. Keywords Edge-Magc Labelg, Complemetary Edge Magc Labelg, 99 Mathematcs Subject Classfcato. O5C78. Itroducto The subject of edge magc labelgs of graphs had ts orgs three decades ago the work of Kotzg ad Rosa [9,0] o what they called magc valuatos of graphs (whch are also commoly kow as edge-magc total labelg see[4]. Iterest these labelgs has bee lately rekdled by a paper o the subject due to Rgel ad Llado[]. Shortly after ths, Eomoto, Llado, Nakamgawa ad Rgel[3] defed a more restrctve form of edge-magc labelgs amely super edge magc labelgs whch Walls[] refers to as strog edge magc labelgs.. Objectve By G(p, q, we deote a graph havg p vertces ad q edge, by V(G ad E(G, the vertex set ad the edge-set of G respectvely. But the vertces ad edges are called the elemets of the graph G. A graph(p, q s sad to have a edge magc wth the magc costat k (whch s depedet o the choce of ay edge uv of G f there exsts a oe oe to oe mappg f:v(g U E(G {,, p+q} such that f(u + f(v +f(uv =k for all uv E (G. If such a labelg exsts, the magc costat k s called valece of f ad G s sad to be edge magc graph. Gve a edge magc f of a graph G (p, q, the Fucto f(x such that f( x = p+q+-f(x for all * Correspodg author: sajay_ole@yahoo.co. (Sajay Roy Publshed ole at Copyrght 0 Scetfc & Academc Publshg. All Rghts Reserved elemets x G s sad to be complemetary to f(x. Two edge magc f ad f of G are equvalet f f =f or f = f. A edge magc f of G s sad to be self complemetary edge magc f f= f Fgure. The edge magc stregth of a graph G s deoted by ems(g ad s defed as the mmum of all costats where the mmum s take over all edge magc labelgs of G. ems(g = m {k: f s a edge magc labelg of G} smlarly the cocept of the complemetary edge magc stregth of a graph s troduced. The complemetary edge magc stregth of G s deoted by cems(g ad s defed as the mmum of all costats k where the mmum s take over all complemetary edge magc labelg of G. cems (G = m { k : f s a complemetary edge magc labelg of G}. 3. Methods I ths paper, the complemetary edge magc stregth of some well kow graphs such as K m, (m,, C ( 3,

2 Amerca Joural of Mathematcs ad Statstcs 0, (3: -6 3 p (, f, B ad G ( where G s bpartte or trpartte are obtaed. The reader s drected to Chartered ad Lesak[] or Hartefeld ad Rgel[8] for all addtoal termology ot provded ths paper. The followg theorem s every useful for the ma results. Theorem A[] A complemetary edge magc graph G satsfes the followg equato p T ( T + qk = + [ dv ] f ( v where f s complemetary edge magc labelg of G ad T=p+q. 4. Results Ma Results I ths secto we proceed to study the complemetary edge magc stregth of K m,, C, f =P +k ad B = K, x K ad G, where G s bpartte or trpartte. Theorem. A complete bpartte graph Km, s complemetary edge magc stregth f m,. Proof: Let G be a complete bpartte graph K m, wth vertex set V parttoed to two subject V ad V where V ={u, u, u m } ad V ={v,v, v } ad let f: VUE {,,,m++m} The, f (v = m++m+- f(v j = mx++ (m+j f(a b j = -m+(m+j + It s easy to see that f exteds to a complemetary edge magc labelg of K m, wth the magc costat k. k = f(v + f (v j + f(a b j = [m++m+-] + [m++ (m+j] + [-m+(m+j +] = m++ = (m++ Complemetary edge magc stregth: Now by Theorem A ( m + + m( m + + m + qk = + ( ( m + + m + j= + [ m + + ( m + j] ( m + + m( m + + m + [ m] k = m + m( ( m + + m + ( Σ ( ( j= + ( m m + + m Σ j ( m + + m + = [ m + + m + m ( ] mm ( + ( ( m + ( + ( m ( m + + k = ( m + + Thus scems (K m, = (m++ The followg theorem s the complemetary edge magc stregth of cycle C for every teger 3. Theorem. The cycle C s complemetary edge magc ad complemetary edge magc stregth s 7 + where s a teger. Proof: Let C be the cycle wth V (C = {v : } ad E (C = {v : v + : } where s take modulo (replacg o by We dscuss the followg cases. Case I: s odd say =m + where m s postve teger. Cosder the fucto f: V(G U E (G {,,} Defed as + f odd f( v ={ + f eve ad, + =, f( vv + ={ = Sce we have U f (V = {,4,6,0,.} U v + = {,3,5.(-} f (v + f (v + f (v v = +(++ = 3+ f (v + f (v + + f (v v + = (+- + (- + (+ =3+ for,, - Thus f s a complemetary edge magc of cycle C wth magc costat 3+ Complemetary edge magc strg: TT ( + qk = + ( dv f ( v (+ k = + f ( v = (+ + { + ( } = (3+ k = 3+ Case. o (mod 4 say = 4l where l s the postve teger, f s defed as + =, 3, l = l,l+ 4l + = l+, l+ 3 4l 3, l f( v = { =,4,l, = 4l 3 = 4l 3+ =,,l, l,l+,4l 3 = l = 4l f ( v v = = { + = 4l = 4l 3 = 4l Clearly U f(v = { 4,6,8,..-4,} U {+, -} U f(v v + = {,3,5,..-, +3,+5,,-3} U {, -} Next we prove that

3 4 Sajay Roy et al.: O Complemetary Edge Magc Labelg of Certa Graphs f(v +f(v + +f(v v + = k for all =,,..,4l Sub case. Let {4, 6, 8.., -4,} U {+,-} The f (v +f (v + +f (v v + = (+-+(--+(3+ =3(+ Sub case. Let = l- the f (v l- +f (v l + f (v l- v l =(-l++(-l+=3(+ Sub case.3 Let {l, l+,, 4l-4} The f (v +f (v + +f (v v + = (-+(-+3+3(+ Sub case.4 Let = l- The f (v 4l- +f (v 4l- +f (v 4l- v 4l- =(++(-+=3(+ Sub case.5 Let I {l+, l+3,..,4l-3}, The f (v +f(v + +f (v v + = (+-+(--+(3+=3(+ Sub case.6 Let {l+, l+3, 4l-3} The f (v +f (v + + f (v v + = (-+(-+(3+=3(+ Sub case.7 Let = 4l-. The f (v 4l- +f (v 4l +f (v 4l- v 4l =(-+(3+(+=3(+ Sub case.8 Let = 4l. The f (v 4l +f(v +f (v 4l v = 3+(+=3(+ Therefore f s a complemetary edge magc of the cycle C wth the magc Costat k = 3 (+ Complemetary edge magc stregth: qk = (+ + f ( v k = ( ( / { 4 + ( } + ( (3+ k = = 3( + Case 3. (mod 4 that s = 4l+. Defe the fucto + / / f =,3, l + / / f = l + 3,l + 5 4l + 6l f = f( v = { l f = l + l, f = 4l + l / f = 4,6, l; l l / f = l + 4,l + 6,4 l; l 4l f = 4l f = 8l 3 f = l + f( vv + = { 8l f = l + 6l f = 4l + 6l f = 4l + 8l 3+ otherwse; l Sce U {f v } = { 4l +, 4l+3,.., 6l+, 6l+,6l+4,6l+5, 8l+4} U {f (v v + } = {,,,4l, 4l+, 6l+3} Next we prove that f (v +f (v + + f (v v + = k for all =,,4l+ Subcase 3. Let = The f (v +f (v + + f (v v + = +(-6l-+(-4l- = (3-5l-=(7+/ Sub case 3. Let = {3,5,..,l-} The f (v +f (v + +f (v v + = (+/ / + {-l-- (+/}+(-8l-3- = (3-5l-= (7+/ Sub case 3.3 Let = l+. The f (v l+ + f (v l+ +f (v l+ v l+ = (-(-l-+(-8l-3=3(-5l-=(7+/ Sub case 3.4 Let = {l+3, l+5,, 4l-} The f (v +f (v + +f (v v + = (-/-/ +{-l--(-/+(-8l-3+ = 6-0l-4 = (3-5l- = (7+/ Sub case 3.5 Let = 4l+. The f (v 4l+ +f (v 4l+ +f (v 4l+,v 4l+ = (-/- (4l+/ +(-l-+(-6l-= 6-0l-4 = (3-5l- = (7+/ Sub case 3.6 Let =. The f (v +f (v +f (v v 3 = (-6l-+(-+(-4l+ = 6-0l-4 = (3-5l- =(7+/ Sub case 3.7 Let = k+. The f (v l+ + f (v l+3 + f (v l+ v l+3 = (-l- +(-l-+(-8l-=(3-5l-= (7+/ Sub case 3.8 Let = 4l+. The f (v 4l+ +f (v +f (v 4l+ v = (-l-+(++(-6l- =(3-5l- =(7+/ Sub case 3.9 Let {4,6, k}. The f (v +f (v + +f (v v + = (-l--/ +(-/+(-8l-3+= (3-5l-= (7+/ Sub case 3.0 Let {l+4, l+6,.4l}. The f (v +f (v + +f (v v + = (-l-/ +(--/+(-8l-3+= (3-5l-=(7+/ Thus f s a complemetary edge magc fucto of the cycle C 4l+ wth the magc costat k =(3-5l-= (7+/ Complemetary edge magc stregth: qk = {(+}/ + f (v = {(+}/+[ (+/+ (-/ x{ (++ (-}] k == 7 + / k = 7+/. Theorem 3. Let G = Р be a regular graph of degree oe. The G s a Complemetary edge magc stregth f ad Oly f s odd. Proof: Sce (3+/ must be ecessarly a teger. Hece t follows that must be odd say = l+ Let u, u, u ad v, v,v ; u v, u v u v be vertces ad edges of G respectvely. Defe f by 3 5l =,, l+ l f( u = 3+, f( v{ 3 3 l = +, l+ 3, l+ Ad 3 4l 3 + =,, l+ f( uv = { 3 6 l 4 + = +, l+ 3, l+ U We have {f u } = {+, +,..3} ad = U {f (u v } = {+,+,.., } = Now we show that k = f (u +f (v +f (u v= (3+- + (3-3l- +(3-6l-4+= 9-9l-3 = 3{(9+}/

4 Amerca Joural of Mathematcs ad Statstcs 0, (3: -6 5 Complemetary edge magc stregth: qk = {3 (3+}/ Therefore, K = {3(3+}/ Theorem 4. The fa f = P +K s a complemetary edge magc for ay postve teger Proof. Let V (f = {u} U { v : } Ad E (f = { uv : } U {v v + : -} Now defe the fucto f: V (G U E(G {,,..,3} as follows 3 f x = u { ( 6 } / 4 f x = v ad f( x = { { 3 5( + 6 } / 4 f x = ux ad 3 f x = v v ad + Observe that f(x+f(y+f(xy=6 for every edge xy E (f Also otce that alteratve type labelg s as follows: {f (v + : o _(-/_ = {3(-+ : _(+/_ } {f (uv : (-/ = {3(-+ :+ _(+/_ } {f (v v + : - = {3(-+ : + _(+/_ } {f (v : (-/ = {3(--: o _(-/_ } {f (uv + : 0 _(-/] _ = {3(--: + _(-/_ -} Ad f (u = 3. Thus all tegers,,,3 are used exactly oce. Hece f s a complemetary edge magc labelg of f wth magc costat (valece 6. Complemetary edge magc stregth : qk = {3(3+}/ + {+3+5(- -6 }/ 4 + 3(--(3-- {6+3+5(- }/ 4 =/[ (- -6(+/ /-5/(- ] (- k =/[4 -+ 5(- +5/-5/(- ] k =/[4 -]/- = 6 f s odd or eve Theorem 5. The book B =K, x K s complemetary edge magc stregth for ay postve teger Proof. Let B be the book defed as follows: V (B = {uv}u{u v : } ad E (B = {uv}u{uu, vv, u v : } Now cosder the fucto f : V(B UE(B {,,..,5+3} Where f x = u f x = v 3 + f x = uv 3 + f x = u ad f( x = { f x = v ad + f x = uu ad + f x = uv ad f x = vv ad 5. Coclusos Fally observe that f s complemetary edge magc labelg of B havg valece 8+6 Complemetary edge magc stregth qk = (5+3(5+4/+(5+3 +(+ (3+-+ (3++ = (5 +35+/ (6+4+ (3+ k = ( / + (6+4+ {(+}/ k = {48 +5+}/{(3+} = 8+6 Ths secto has a tool that allows us to geerate fte classes of dscoected complemetary edge magc - partte graphs wth relatve case where =,or 3. Prevously o such a techque was avalable for graphs wth those classes sec[6] except [5]. Theorem 6. If G s a complemetary edge magc bpartte or trpartte graph ad s odd the G ( s complemetary edge magc. Proof. Assume that 3. If (p,q graph G s a complemetary edge magc bpartte or trpartte graph wth partte sets A, B, C [Let C=Φ f G s bpartte ] the let E(G = AB U AC U BC where the juxta-posto of two partte sets deotes the set of edges betwee those two sets. Let f :V (GUE(G {,,,p+q} to be a arbtrary complemetary edge magc labelg of G. The X G to be the graph wth vertex set V(X= U (A B UA C UB C where x Y for = f ad oly f x Y (Y s oe of the sets A,B,C,AB,AC,BC Cosder the labelg g : V(XUE(X {,,.. (p+q} such that f ( x + f x CUAB f ( x + f x AUBC ( / f( x = { f ( x + f x AUBC ( + / f ( x / + / + f x BUAC ( / f ( x + / + / f x BUAC ( + / Clearly g s a complemetary edge magc labelg of X wth valece g(u+g(v+g(uv=(p+q-3(-/ for each edge uv E(X where k f s the valece of f. Next to observe that g(v(hue(h={,,(p+q s spread by the fucto g to the etrety of ts rage. I the above theorem ca ot be eve for Kotzg ad Rosa [9] have show that the forest P s edge magc f ad oly s odd Immedately the corollary follows from above Theorem Corollary: If s odd ad m> the -regular graph C m s complemetary edge magc. Poof. Kotzg ad Rosa have show that all cycles are edge magc. But a alteratve labelg of eve cycles ca be foud [7].

5 6 Sajay Roy et al.: O Complemetary Edge Magc Labelg of Certa Graphs REFERENCES [] D. Craft ad E. H Tesar O a questo by Erdos about edge magc graph.dscrete Math. 07( [] G. Chartrad ad L. Lesak, Graphs ad Dgraphs,Wadsworth ad Books/cole, Advaced Books ad software Moterey, C.A(986 [3] H. Eomoto, A Llado, T. Nakamgawa ad G. Rgel, Super edge magc graphs SUT J. Math. 34(998 [4] R.M Fgueroa Ceteo, R. Ichshma ad F. A Mutaer Batle, Elargg the classes of edge magc regular graphs, (pre-prt [5] R. M. Fgueroa Ceteo, R Ichshma ad F. A Mutaer- Batle, O edge magc labelg of certa dsjot uos of graphs, (pre-prt [6] J. A Galla, A dyamc survey of graph labelg, Electro J. Comb. S (00 #D56 [7] R. D Godbold ad P. Stater, All cycle are edge magc Bull Ist. Comb. Appl. ( [8] N.Harts feld ad G. Rgel, Pearls Graph Theory, A comprehesve Itroducto, Academc Press, Sa Dego, (994 [9] A. Kotzg ad A. Rosa, Magc valuatos of fte graphs, Caad Math. Bull. 3( [0] A. Kotzg ad A. Rosa, Magc valuatos of complete graphs, Publcatos du Cetre de Recherches Mathematques Uverste de Motreal, 75(97 [] G. Rgel ad A Llado, Aother tree cojecture, Bull. Ist. Comb. Appl. 8( [] W. D Walls, Magc Graphs, Brkhauser Bosto (00