On Face Bimagic Labeling of Graphs

Transcription

1 IOSR Joural of Mathematcs (IOSR-JM) e-issn: , p-issn: X Volume 1, Issue 6 Ver VI (Nov - Dec016), PP wwwosrouralsor O Face Bmac Label of Graphs Mohammed Al Ahmed 1,, J Baskar Babuee 1 1 (Departmet of Mathematcs, Aa Uversty, Chea , Ida) (Departmet of Mathematcs Collee of Educato for pure Sceces Ib AL Hatham, Uversty of Bahdad, Iraq) Abstract: Let G ( V, E, F) be a smple, fte, plae raph wth V (G) umber of vertces, E (G) umber of edes ad F (G) umber of faces A label of type (1,1,0) s a bectve mapp from the vertex set ad ede set of G to the set {1,,, V (G) + E (G) } Moreover, the label s called super f the vertces are labeled wth the smallest umbers The weht of a face uder the label s the sum of labels of edes ad vertces surroud that face I ths paper we study face bmac label of type (1,1,0) for wheels, cylders ad dsot uo of m copes of prsm raphs Keywords: Wheel, cylders, prsm, face bmac label I Itroducto I ths paper we cosder fte, smple, udrected ad plae raphs If G s a plae raph, wth the vertex set V (G), ede set E (G) ad face set F (G) A label of type (1,1,0) asss labels from the set {1,,, V (G) + E (G) } to the vertces ad edes of plae raph G such a way that each vertex ad ede receve exactly oe label ad each umber s used exactly oce as a label Ths label s called mac f for every postve teer s the set of s-sded faces have the same wehts The oto of mac label of plae raphs was defed by Ko-We Lh [0], ad some of the platoc famly are ve The mac label of type (1,1,1) for fas, plaar bpyramds ad ladders, s ve [6] by Bača, who also proves that rds, hexaoal lattces, Mobus ladders ad P P3 have mac label of type (1,1,1) [7,8,9,10] respectvely Bača proves the cylders C Pm have mac label of type (1,1,0) whe m, 3, 4 [11] Kathresa ad Gokulakrsha [19] provded mac label of type (1,1,1) for the famles of plaar raphs wth 3-sded faces, 5-sded faces,6-sded faces, ad oe exteral fte face Al, Hussa, Ahmed, ad Mller [3] study mac label of type (1,1,1) for wheels ad subdvded wheels Bača [1, 13, 14, 15, 9, 16] ad Bača ad Holläder [17] ave mac labell of type (1,1,1) ad type (1,1,0) for certa classes of covex polytopes I 004 JB Babuee [4] troduced the cocept of ede bmac total label ad studed for some famles of raphs [5] Abecto from V( G) E( G) to the set {1,,, V( G) E( G) } s called ede bmac total label f the ede wehts are ether equal to a costat k1 or to a costat k, where the ede weht of a ede s the sum of the ede labels ad the labels of ts ed vertces More ormal lterature about ede bmac s avalable [18] II Ma Results I [] authors troduced the cocept of face bmac label Defto 1: Let G = (V (G), E (G), F (G)) be a smple, fte, coected plae raph wth the vertex set V (G), the ede set E (G) ad the face set F (G) A becto from V( G) E( G) to the set {1,,, V( G) E( G) } s called face bmac f for every postve teer s the weht of every s-sded face s equal ether to ks or to t s We allow dfferet umbers k, t for dfferet s Moreover, f for every postve teer s, the umbers of s- s s sded faces wth wehts k s ad t s dffer by at most oe, the ths label s called equtable face bmac Ths paper deals frst, wth the ope problem that every eve wheel W s (1,1,0) face mac label [3], the problem s stll ope, however we wll prove that every eve wheel W s (1,1,0) super equtable face bmac label, ad the we wll study super face bmac label for cylders ad dsot uo of m copes of prsm raphs The ext theorem deals wth super equtable face bmac label of type (1,1,0) for wheel raphs Theorem : The wheel raph W admts super equtable face bmac label of type (1,1,0) for every eve, 4 Proof Let the vertex set ad the ede set of the wheel raph W be V W v 1,, u ( ) :,, E( W ) DOI: / wwwosrouralsor 1 Pae

2 O Face Bmac Label of Graphs { vv 1, vv1 : 1,,, 1} { uv : 1,,, } The set of faces s F( W) { f : 1,,, }, where the boudares of the faces f for 1 are defed as follows: for1 1, the boudares of the faces are uvv 1, ad for the boudary of the face f s uuv 1 For s eve, 4, : V( W ) E( W ) 1,,,31 such that u ( ) 1, 1 for we defe a bectve mapp v 1,,,, 1 for 1,3,, 1, u v for,4,,, 3 for 1,,,, vv for 1,,, 1, vv1 1 For the face wehts of faces f for 1,,,, we et w ( f ) ( u) ( v ) ( v ) ( uv ) ( v v ) ( v u) for 1,,,, for 1,,, 1, w ( f) ( u) ( v ) ( v1 ) ( uv ) ( vv1 ) ( v1u ) Hece the wheel raphw, admts a super equtable face bmac label of type (1,1,0) wth two mac costats, respectvely Theorem 3: For m, the cylders C Pm have super face bmac label of type 1,1,0 for every eve, 6 Proof Let the vertex set ad the ede set of the cylders C Pm be V( C P m ) { v : 1,,,, 1,,, m}, E( C P ) { v v, v v : 1,,, 1, 1,,, m} { v v : 1,,,, 1,,, m1} The set of m faces s F( C Pm ) { f : 1,,,, 1,,, m1} { fer, fexteral }, where the boudares of the faces f for 1,,, 1, 1,,, m 1 are defed as v, v 1v 1 1v 1 ad the boudary of the face f, s v v v v For eve,, 1,,, m, m we defe a bectve mapp : V( C P ) E( C P 1,,, 3m for 1,,,, f s odd v 1 for 1,,,, f s eve 3m for 1,,,, vv 1 3m 1 for 1,,, 1, v v1 3m 1, v v m 1 for 1,,, 1 For the wehts of faces f for 1,,, 1, 1,,, m 1, we et ), m m such that for DOI: / wwwosrouralsor Pae

3 O Face Bmac Label of Graphs w ( f ) ( v ) ( v ) ( v ) ( v ) ( v v ) ( v v ) ( v v ) ( v v ) ( ) ( ( 1) 1 ) ( ( 1) 1 ( 1)) ( 1) (m 1 ) (3 m ( 1) ) (m 1 ( 1)) (3 m ) for 1,,,,f s odd ( 1 ) ( ( 1) ) ( ( 1) 1) ( 1 ( 1)) (m 1 ) (3m ( 1) ) (m 1 ( 1)) (3 m ) for 1,,,,f s eve ( ) ( ( 1) 1 ) ( ( 1) 1 ( 1)) ( 1) (m 1 ) (3m ( 1) 1) (m 1 ( 1)) (3m 1) for 1,,, 1,f s odd ( 1 ) ( ( 1) ) ( ( 1) 1) ( 1 ( 1)) (m 1 ) (3m ( 1) 1) (m 1 ( 1)) (3m 1) for 1,,, 1, f s eve 10m 3 3 for 1,,,, 10m 3 5 for 1,,, 1, w ( f ) ( v ) ( v ) ( v ) ( v ) ( v v ) ( v v ) ( v v ) ( v v ) ( ) ( ( 1) 1 ) ( ( 1)) ( 1) (m 1 ) (3m 1 ( 1) ) ( m ) (3m 1 ) f s odd ( 1 ) ( ( 1)) ( ( 1) 1) ( ) (m 1 ) (3m 1 ( 1) ) ( m ) (3m 1 ) f s eve 10m 3 5 for 1,,, m 1, 1 w ( f ) ( v ) ( v v ) ( v v ) er / 1 (3m ) ( (3m 1 )) ( 3m 1 ) 3 m 1 1 ( /) 1 1 w ( f ) ( v ) ( v v ) ( v v ) exteral m m 1m m 1m 1 1 / 1 ( m ) ( m ) (m 1 ) 1 1 ( /) 1 ( m 1 ) f m odd / 1 ( m 1 ) ( m ) (m 1 ) 1 1 ( /) 1 ( m 1 ) f m eve 3 m for both m odd ad eve Now, by swapp the label of the ede v m 1vm wth the label of the vertex v m, we ca easly chae the weht of exteral face from 3 m to 3 m 1, f m s odd, ad to 3 m, f m s eve, Ths swapp does ot have ay effect o the wehts of 4-sded faces, hece the cylders C Pm admts a super face bmac label of type (1,1,0) wth two mac costats 10m 3 3, 10m 3 5, for 4-sded faces, ad 3 m, 3 m 1, f m s odd, -sded faces respectvely 3 m, 3 m, f m s eve for DOI: / wwwosrouralsor 3 Pae

4 O Face Bmac Label of Graphs Corollary 4: For m, the cylders C4 Pm 1,1,0 Proof Let the vertex set, ede set ad face set of cylders C4 Pm as were defed the prevous theorem have super face bmac label of type v4 m v1 m v4 m-1 f4 m-1 v1 m-1 v43 v13 v4 f4 f41 v41 v11 v1 f3 m-1 f3 f31 f11 f1 f1 m-1 v3 v31 v1 f1 f v v33 v3 v3 m-1 v m-1 f m-1 v3 m F 1 (Cylders C4 Pm ) v m m we defe a bectve mapp : V( C P ) E( C P ) 1,,,1m 4 For, 1,,, m, 4 4 for 1,,3,4, f s odd v 4 1 for 1,,3,4, f s eve 8m 4 for 1,, 3, f s odd vv 1 8m 4 1 for 1,, 3, f s eve 8m 4 4 f s odd v4v1 8m 4 1 f s eve Ad for 1,,, m 1, vv 1 1m 1 4 for 1,, 3, 4 4 m 4 For the wehts of faces f for 1,,3,4, 1,,, m 1, we et w ( f ) ( v ) ( v ) ( v ) ( v ) ( v v ) ( v v ) ( v v ) ( v v ) m such that for (4 4 ) (4( 1) 1 ) (4( 1) 1 ( 1)) (4 4 1) (1m 1 4 ) (8m 4( 1) 1 ) (1m 1 ( 1) 4 ) (8m 4 ) for 1,, 3, f s odd (4 1 ) (4( 1) 4 ) (4( 1) 4 ( 1)) (4 1 ( 1)) (1m 1 4 ) (8m 4( 1) ) (1m 1 ( 1) 4 ) (8m 4 1) for 1,,3, f s eve 40m for 1,, 3, f s odd =40 m for 1,, 3, 1,,, m 40m for 1,, 3, f s eve w ( f ) ( v ) ( v ) ( v ) ( v ) ( v v ) ( v v ) ( v v ) ( v v ) (4 ) (4( 1) 3) (4( 1)) (4 3) (1 m 3 4 ) (8m 4( 1) 1) (1m 4 ) (8m 4 4) f s odd (4 3) (4( 1)) (4 1) ( 4 ) ( 1m 3 4 ) (8m 4( 1) 4) (1m 4 ) (8m 4 1) f s eve 40 m for 1,,, m 1, DOI: / wwwosrouralsor 4 Pae

5 O Face Bmac Label of Graphs w ( f ) ( v ) ( v v ) ( v v ) er (8m 4 ) 8m 3m w ( f ) ( v ) ( v v ) ( v v ) exteral m m 1 m 4 m 1m 1 1 (4m 4 ) (4 m ) (4m 4) f m odd 1 1 (4m 1 ) (4m 1) ( 4m 1) f m eve 1 1 3m 4 for m odd ad eve Hece the cylders raph C 4 P m, admts a super face bmac label of type (1,1,0) wth two mac costats 40 m, 3m4 respectvely Theorem 5 For every eve, 6, m, the dsot uo of m copes of a prsm raph ( C P ) admts super face bmac label of type 1,1,0 Proof Let the vertex set ad ede set of a raph m C P ) be V( m( C P )) { v : 1,,,, 1,, m}, ( E( m( C P )) { v v : 1,,, 1, 1,,, 1, 1,,, m} { v v : 1,,,, 1,,, m} 1 { vv1, vv 1} The set of faces s m C P er exteral F ( ( )) { f : 1,,,, 1,,, m} { f, f : 1,,, m }, where the boudares of the faces f for 1,, 1, 1,,, m, are v vv 1v 1 ad the boudary of the face f s vv v 1v1 For eve, 6, m, : V( m( C P )) E( m( C P )) 1,,,5m such that we defe a bectve mapp ( v ) for 1,,,, 1,,, m, ( v ) m 1 for 1,,,, 1,,, m, ( v v ) 3m 1 for 1,,,, 1,,, m, 4m for 1,,,, 1,,, m, ( vv 1) 4m 1 for 1,,, 1, 1,,, m, 5m for 1,,,, 1,,, m, ( v v 1) 5m 1 for 1,,, 1, 1,,, m, ( vv1 ) 4m 1, ( vv 1) 5m 1 For the wehts of faces f for 1,,, 1, 1,,, m, we et w ( f ) ( v ) ( v ) ( v ) ( v ) ( v v ) ( v v ) ( v v ) ( v v ) ( ) ( m 1 ) ( m ) ( 1) (3m 1 ) (4 m ) (3 m ) (5 m ) for 1,,,, 1,,, m, ( ) ( m 1 ) ( m ) ( 1) (3m 1 ) (4m 1 ) (3m ) (5m 1 ) for 1,,, 1, 1,,, m, DOI: / wwwosrouralsor 5 Pae

6 O Face Bmac Label of Graphs 17m 3 for 1,,,, 1,,, m, 17m 5 for 1,,, 1, 1,,, m, w ( f ) ( v ) ( v ) ( v ) ( v ) ( v v ) ( v v ) ( v v ) ( v v ) ( ) ( m 1 ) ( m ) ( 1) (3m 1) (5m 1 ) (3 m ) (4m 1 ) 17m 5 for, 1,,, m, For the wehts of -sded faces, we et 1 er w ( f ) ( v ) ( v v ) ( v v ) / 1 ( ) (5 m ) (5m 1 ) (4m 1 ) 1 1 ( /) 1 5m m for 1,,, m, 1 exteral w ( f ) ( v ) ( v v ) ( v v ) / 1 ( m 1 ) (4 m ) (4m 1 ) (5m 1 ) 1 1 ( /) 1 5m m for 1,,, m, Observe that the dfferece betwee the vertex label v ( /) ad the vertex label v ( 3 /) s m 1, so that f we swap the vertex label ( v/) wth the vertex label v ( 3 /), the the wehts of -sded faces wll be w ( f ) 5m 1 for 1,,, m, er w f m m ( exteral ) 5 1 for 1,,,, Ths swapp does ot have ay effect o the wehts of 4-sded faces, hece the raph m ( C ) P admts super face bmac label of type (1,1,0) wth two mac costats 17m 3, 17m 5, for 4-sded faces ad 5m 1, 5m 1, for -sded faces respectvely Corollary 6 For 4, m the dsot uo of m copes of a prsm raph ( C4 P) admts super face bmac label of type 1,1,0 Proof The vertex set, ede set ad face set as were defed the prevous theorem For = 4, m, we defe : V( m( C P )) E( m( C P )) 1,,,0m, such that a bectve mapp 4 4 ( v ) 4 4 for 1,,3,4, 1,,, m, ( v ) 4m 4 1 for 1,,3,4, 1,,, m, 4 ( v v ) 1m 5 4 for 1,,3,4, 1,,, m, 4 ( v v ) 16m 4 for 1,,3, 1,,, m, 4 41 ( v v ) 0m 4 1 for 1 ( v v ) 0m 1 4 for 1,,, m, 4 1 ( v v ) 16m 4 4 for 1,,, m 8 5 1,,3, 1,,, m, It s easy to observe that, the wehts of the faces f, for 1,,3,4, 1,,, m, s 68m 4 ad the wehts of the er ad exteral faces are 80m 4, for 1,,, m Hece the raph m( C4 P ) admts super face bmac label of type (1,1,0) for every m DOI: / wwwosrouralsor 6 Pae

7 O Face Bmac Label of Graphs III Cocluso I the foreo secto we vestated the exstece of face bmac label of type 1,1,0 for certa famles of raphs, maly for wheels, cylders C Pm ad dsot uo of m copes of prsm raph Ackowledemets Mohammed Al Ahmed ratefully ackowledes the facal support from Bahdad Uversty, Mstry of Hher Educato ad Scetfc Research, Iraq ad to Ida Coucl for Cultural Relatos for support to peruse research Aa Uversty, Chea, Ida Refereces [1] A Ahmad, M Bača, M Lascsáková, ad A Semačová-Feňovčíková, Super mac ad atmac labels of dsot uo of plae raphs, Scece Iteratoal, 4, 01, 1-5 [] MA Ahmed, ad JB Babuee, Bmac label for stro face plae raphs, Iteratoal Coferece o Mathematcal Computer Eeer, VIT Uversty, Chea, Ida, 015 [3] K Al, M Hussa, A Ahmad ad M Mller, Mac Labels of type (a,b,c) of famles of wheels, Math Computer Sc, 7, 013, [4] JB Babuee, Bmac label path raphs, The Mathematcs Educato, 38, 004, 1-16 [5] JB Babuee, O ede bmac label, J Comb If Syst Sc, 8, 004, [6] M Bača, O mac ad cosecutve labels for the specal classes of plae raphs, Utltas Math, 3, 1987, [7] M Bača, O mac labels of rd raphs, Ars Comb, 33, 199, [8] M Bača, O mac labels of hoeycomb, Dscrete Math, 105, 199, [9] M Bača, O mac labels of Mӧbus ladders, J Frakl Ist, 36, 1989, [10] M Bača, O mac labels of type (1, 1, 1) for three classes of plae raphs, Math Slovaca, 39, 1989, [11] M Bača, O mac labels of m-prsms, Math Slovaca, 40, 1990, [1] M Bača, Labels of m-atprsms, Ars Comb, 8, 1989, 4-45 [13] M Bača, Labels of two classes of covex polytopes, Utltas Math, 34, 1988, 4-31 [14] M Bača, Labels of two classes of plae raphs, Acta Math Appl Sca, 9, 1993, 8-87 [15] M Bača, O mac labels of type (1, 1, 1) for the specal class of plae raphs, J Frakl Ist, 39, 199, [16] M Bača, O mac labells of covex polytopes, A Dsc Math, 51, 199, [17] M Bača ad I Holläder, Labels of a certa class of covex polytopes, J Frakl Ist, 39, 199, [18] J Galla, A dyamc survey of raph label, The Electroc Joural of Combatorcs, 015, #DS6 [19] K Kathresa ad R Gaesa, A label problem o the plae raphs P ab,, Ars Comb, 73, 004, [0] KW Lh, O mac ad cosecutve labels of plae raphs, Utltas Math, 4, 1983, DOI: / wwwosrouralsor 7 Pae