Expanding Super Edge-Magic Graphs

Transcription

1 PROC. ITB Sas & Tek. Vol. 36 A, No., 00, Exadg Suer Edge-Magc Grahs E. T. Baskoro & Y. M. Cholly, Deartet of Matheatcs, Isttut Tekolog Badug Jl. Gaesa 0 Badug 03, Idoesa Eals : {ebaskoro,yus}@ds.ath.tb.ac.d Deartet of Matheatcs, Uverstas Muhaadyah Malag Jl. Tlogoas 6 Malag 65, Idoesa Eal : yus@u.ac.d Abstract. For a grah G, wth the vertex set V(G) ad the edge set E(G) a edgeagc total labelg s a becto f fro V ( G) E( G) to the set of tegers {,,, V ( G) E( G) } wth the roerty that f ( u ) f ( v) f ( uv) = k for each uv E(G) ad for a fxed teger k. A edge-agc total labelg f s called suer edge-agc total labelg f f ( V ( G)) = {,,, V ( G) } ad f ( E( G)) = { V ( G), V ( G ),, V ( G) E( G) }. I ths aer we costruct the exaded suer edge-agc total grahs fro cycles C, geeralzed Peterse grahs ad geeralzed rss. Keywords: Edge-agc; suer edge-agc; agc-su. Itroducto All grahs cosdered here are fte, udrected ad sle. As usual, the vertex set ad edge set wll be deoted V (G) ad E(G), resectvely. The sybol A wll be deote the cardalty of the set A. Other terologes or otatos ot defed here ca be foud [,7,5]. Edge-agc total labelgs were troduced by Kotzg ad Rosa [8] as follow. A edge-agc total labelg o G s a becto f fro V ( G) E( G) oto {,,, V ( G) E( G) } wth the roerty that, gve ay edge uv, f ( u) f ( v) f ( uv) = k for soe costa k. It wll be coveet to call f ( u) f ( v) f ( uv) the edge su of uv ad k the agc su of f. A grah s called edge-agc total f t adts ay edge-agc total labelg. Ths work was suorted by Hbah Bersag XII DP3M-DIKTI, DIP Nuber : 00/XXIII//--/00.

2 8 E. T. Baskoro & Y. M. Cholly Kotzg ad Rosa [9] showed that o colete grah K wth > 6 s edgeagc total ad ether s K, ad edge-agc total labelgs for K 3, K5 ad for all feasble values of k, are descrbed []. K 6 I [8] t s roved that every cycle C, every caterllar (a grah derved fro a ath by hagg ay uber of edat vertces fro vertces of the ath) ad every colete bartte grah K (for ay ad ) are edge-agc total., Walls et.al. [] showed that all aths P ad all -sus (a cycle C wth a addtoal edge teratg a vertex of degree attached to each vertex of the cycle) are edge-agc total. It was show [6] that the Cartesa roduct C adts a edge-agc total labelg for odd. P It s coectured that all trees are edge-agc total [8] ad all wheels W are edge-agc total wheever 3 (od ) []. Eooto et.al. [] showed that the coectures are true for all trees wth less tha 6 vertces ad wheels W for 30. Phls et.al. [] solved the coecture artally by showg that a wheel W, 0 or (od ), s edge-agc total. Sla et.al [3] showed that for 6 (od 8) every wheel W has a edge-agc total labelg. A edge-agc total labelg f s called suer edge-agc total f f ( V ( G)) = {,,, V ( G) } ad f ( E( G)) = { V ( G), V ( G),, V ( G) E( G) }. Eooto et.al. [] roved that the colete bartte grahs K, suer edge-agc total f ad oly f = or =. They also roved the colete grahs K s suer edge-agc f ad oly f =, or 3. I ths aer we wll costruct the suer edge-agc total grahs by hagg ay uber of edat vertces fro vertces of the cycles, geeralzed rss ad geeralzed Peterse grahs. s Results For 3 ad we deote by C vertces ad edges to oe vertex of cycles C (say A a grah whch s obtaed by addg v v ). The vertex set ad the edge set of C A are V ( C A ) = { : } { u : } ad E( A ) = { v : } { v } v { v : }. C v u

3 Exadg Suer Edge-Magc Grahs 9 Let (, ) su be a grah derved fro a cycle C, 3, by hagg edat vertces fro all vertces of the cycle. Let us deote the vertex set of (, ) su by V ((, ) su) = { v : } { u, :, } ad the edge set by E((, ) su) = { vv : } { v v} { vu, :, }. Observe that V ((, ) su) = E((, ) su ) = ( ). The cycle C, 3, s suer edge-agc total f ad oly f s odd (see []). Now, we shall vestgate suer edge-agc total labelgs for grahs of C A ad (, ) su whch are exaded fro a cycle C. Defe a vertex labelg ad a edge labelg f of f C A as follows, f s odd, f ( ) = v f s eve, f ( u ) = for, f ( vv ) = ( ) for, f ( v v) =, ( v u ) = for. f C A s suer edge- Theore. If s odd, 3 ad, the grah agc total. Proof. It s easy to verfy that the values of of f are f are,,, ad the values,,, ad furtherore the coo edge 5 3 su s k =. ; (, ) su s suer edge- Theore. If s odd, 3 ad, the grah agc total. Proof. Label the vertces ad the edges of (, ) su the followg way. f v ) = f ( v ) for, 3( 3 ( u, ) = for, 3( u, ) = ( ) for ( vv ) = ( ) f f ad, f for,

4 0 E. T. Baskoro & Y. M. Cholly f ( vv) =, ( ) f( v u, ) = ( ) ( ) f = ad, f s odd, 3 ad, f s eve, ad. We ca see that the vertces of (, ) su are labeled by values,,, ( ) ad the edges are labeled by ( ), ( ),, ( ). 3 ; Furtherore, all edges have the sae agc uber k = ( ). A geeralzed Peterse grah P(, ), 3 ad, cossts of a outer -cycle v, v,,v a set of sokes vz,, ad er edges z z,, wth dces take odulo. For 5, = ad, we deote by P (,) obtaed by addg vertces ad edges to oe vertex of Hece, E(P(,)) V P(,) A ) = V ( P(,)) {u : } ad E( P(,) A ) = ( { v u : }. A for a grah whch s P (,), say. Let P (,, ) be a grah derved fro P(,), 5, by hagg edat vertces fro all vertces v, o P(,). The the vertex set of P (,, ) s V ( P (,, )) = V ( P(,)) :, } ad the edge f { u, set s E( P(,, )) = E( P(,)) v u :, }. {, I [] t s roved that geeralzed Peterse grahs P(,) are edge-agc total. Fukuch [6] showed that P(,) are suer edge-agc total. v Theore 3. If s odd, 5 ad, the the grah suer edge-agc total labelg. P (,) A has a Proof. Cosder a becto, f : V ( P(,) A ) {,,, } where, 5 = f s eve,, f ( ) 5 v 3 f s odd,,

5 Exadg Suer Edge-Magc Grahs f (od ), f (od ), ( ) = 3 f5 z f 3 (od ), f 0 (od ), f5 ( u ) = for. We ca observe that uder the labelg f, f ( v ) f ( v ) : } = { 5 5 { 5 5 } = { 3 : } ad { f z ) f ( ) : : } wth 5( 5 z dces take odulo. Moreover, { f5( v ) f 5(z ): } = { : } 7 { 5 5 u { f5( z ) f5( z : { 5 f5 u ad f ( v ) f ( ) : } = { : }. The eleets of the set { f 5 ( v ) f5( v ) : } ) } { f 5 ( v ) f5( z ) : } f ( v ) ( ) : } for a arthetc sequece, 7 7,,,,, 7. We are able to arrage the values,,, 5 to the edges of P (,) A such way that the resultg labelg s total ad every edge xy E( P(,) A ), f 5 (x) 3 f 5( y) f5( xy) =. Thus we arrve at the desred result. ; Theore. If s odd, 5 ad, the the grah P(,, ) has a suer edge-agc total labelg. Proof. Defe a becto, f : V ( P(,, )) {,,, ( )} as follows, 6 f v ) = f ( v ) ad f z ) = f ( z ) for, 6( 5 6 ( u, ) = ( ) ( u, ) ( ) 6( 5 f for, f6 = for ad. We ca see that uder the vertex labelg edges 7 f the values f x) f ( ) of all xy E( P(,, )) costtute a arthetc sequece, 7. 7,,, 6 6( 6 y,, If we colete the edge labelg wth the cosecutve values the set { ( ), ( ), ( ) 3,, 5 } the we ca obta total labelg where f ( ) f6( y) f 6 ( xy) = 3 for every edge xy E( P(,, )). ; 6 x

6 E. T. Baskoro & Y. M. Cholly I the sequel we shall cosder a grah of a geeralzed rs whch ca be defed as the Cartesa roduct C of a cycle o vertces wth a ath o vertces. Let V ( C P ) = { v, : ad k } E( C k ( C k {, k, k be the vertex set ad P ) = v v : ad k } { v, kv, k : ad } be the edge set, where s take odulo. For 3, ad we wll cosder a grah ( C P ), ) P A = vertex of C P, say v, (resectvely to all vertces v,, of C P ). Thus V (( C P ) A ) = V ( C P ) { u : }, A (resectvely a grah ) whch s obtaed by addg vertces ad edges to oe V (( C P ) A ) = V ( C ) { u, = :, }, E (( C ) A ) = E( C ) { v, u : }, ad E(( C P ) A ) = E( C ) { v, u, = :, Fgueroa-Ceteo et.al. [5] showe that the geeralzed rs edge-agc f s odd ad. }. C P s suer The ext two theores show suer edge-agc total labelgs of grahs ( C P ) A ad ( C P ) A. = Theore 5. If s odd, 3, ad, the the grah has a suer edge-agc total labelg. ( C P ) A Proof. If s eve,, k,, the we costruct a vertex labelg the followg way, f 7

7 Exadg Suer Edge-Magc Grahs 3 ( k ) f s odd ad s odd, k k f s eve ad k s odd, f ( ) = 7 v, k k f s odd ad k s eve, ( k ) f s eve ad k s eve, f7 ( u ) = for. If s odd, 3, k,, the we defe a vertex labelg f 8 as follows, ( k ) ( k ) f8( v, k ) = k ( k ) ( k ) f8 ( u ) = for. f f f = ad k s eve, f f s odd ad k s odd, s eve ad k s odd ad k s eve, s eve ad k s odd, s eve, It s easy to verfy that for each edge xy E(( C ) A ) the values f x) f ( ) ad f x) f ( ) for a arthetc sequece, 7( 7 y 8( 8 y,, 3,,., Let f 9 be a becto fro E (( C ) A ) oto {,,, }. We ca cobe the vertex labelg f7 (or f8 ) ad the edge labelg f 9 such that the resultg labelg s total ad the edge su for each edge 3 xy E(( C ) A ) s equal to 3. ; Theore 6. If s odd, 3,, ad, the the grah C P ) A = has a suer edge-agc total labelg. Proof. Defe vertex labelg ad f such that : f0 f v ) = f ( v ) f s eve, k,, 0(, k 7, k f v ) = f ( v ) f s odd, k,, (, k 8, k f 0 ( u,, ) = f( u ) = ( ) for, (

8 E. T. Baskoro & Y. M. Cholly f 0 (, u, ) = f( u ) = ( ) for ad. We ca see that vertces of ( C P ) A are labeled by values,, 3,, ( ) t {0,}. = ad ft ( x) ft (y) for all edges xy ( C P ) A ad costtute a arthetc sequece We ca colete the edge labelg of ( C P ) = =,,, A wth values the set { ( ), ( ),, ( 3 )} cosecutvely such that the coo edge su s k = 3 3. ( C P ) = A Thus the total labelg of s suer edge-agc ad the theore s roved. ; Refereces. Bača, M., Cosecutve-agc labelg of geeralzed Peterse grahs, Utltas Math. 58 (000), Bača, M., MacDogall, J. A., Mller, M., Sla & Walls, W. D., Survey of certa valuatos of grahs, Dscussoes Math. Grah Theory 0 (000), Bača, M., L, Y., Mler, M. & Sautak, R., New costructos of agc ad atagc grahs labelgs, Utltas Math. 60 (00), Eooto, H., Lladó, A. S., Nakagawa, T. & Rgel, G., Suer edgeagc grahs, SUT J. Math. Vol. 3 (998), Fgueroa-Ceteo, R. M., Ichsha, R. & Mutaer-Batle, F. A., The lace of suer edge-agc labelgs aog other classes of labelgs, Dscrete Math. 3 (00), Fukuch, Y., Edge-agc labelgs of geeralzed Peterse grahs P(,), Ars Cob. 59 (00), Hartsfeld, N. & Rgel, G., Pearls Grah Theory, Acadec Press, New York, d Edto, Kotzg, A. & Rosa, A., Magc valuatos of fte grahs, Caad. Math. Bull. 3 (970),. 5-6.

9 Exadg Suer Edge-Magc Grahs 5 9. Kotzg, A. & Rosa, A, Magc valuatos of colete grahs, Publ. CRM 75 (97). 0. Mller, M. & Bača, M., Atagc valuatos of geeralzed Peterse grahs, Australasa J. Cob. (000), Ngurah, A. A. G. & Baskoro, E. T., O agc ad atagc total labelg of geeralzed Peterse grahs, Utltas Math. 63 (003), Phlls, N. C. K., Rees, R. S. & Walls, W. D., Edge-agc total labelg of wheels, Bull. ICA 3 (00), Sla, Bača, M., L, Y., Mller, M. & Sautak, R., Edge-agc total labelgs of wheels, fas ad fresh grahs, Bull. ICA 35 (00), Walls, W.D., Baskoro, E. T., Mller, M. & Sla, Edge-agc total labelgs, Australasa J. Cob. (000), Walls, W. D., Magc Grahs, Brkhäuser, Bosto-Basel-Berl, Waya, K. & Baskoro, E. T., Edge-agc labelgs of a roduct of two grahs, Proc. Sear MIPA, ITB Badug (000),. 0-.