Mean Cordial Labeling of Certain Graphs

Transcription

1 J Comp & Math Sc Vol4 (4), 74-8 (03) Mea Cordal Labelg o Certa Graphs ALBERT WILLIAM, INDRA RAJASINGH ad S ROY Departmet o Mathematcs, Loyola College, Chea, INDIA School o Advaced Sceces, VIT Uversty, Chea, INDIA (Receved o: August 0, 03) ABSTRACT Let G be a graph wth p vertces ad q edges A vertex labelg : V ( G) {0,,} s sad to be a mea cordal labelg o G t duces a edge labelg * gve by ( u) + ( v) suchthat v ( ) v ( j ) ad e ( ) e ( j ),, j {0,, }, where v ( r) ad e ( r) deote the umber o vertces ad edges respectvely labeled wth r( r = 0,, ) A graph G s sad to be a mea cordal graph t admts a mea cordal labelg I ths paper, we establsh the mea cordal labelg o caterpllar, S (P K ), S( B, ), S( P P ) ad baaa tree Keywords: Mea cordal labelg, Mea cordal graph, Caterpllar, Baaa tree, Path baaa tree INTRODUCTION All graphs ths paper are te, smple ad udrected The vertex set ad edge set o a graph are deoted by V ( G) ad E( G ) respectvely The cocept o cordal labelg was troduced by Caht Let be a ucto rom V ( G ) to {0,} ad let each edge u v be assged the label ( u) ( v) The s a cordal labelg o G the umber o vertces labeled wth 0 ad the umber o vertces labeled wth der by at most, ad the umber o edges labeled wth 0 ad the umber o edges Joural o Computer ad Mathematcal Sceces Vol 4, Issue 4, 3 August, 03 Pages (0-3)

2 75 Albert Wllam, et al, J Comp & Math Sc Vol4 (4), 74-8 (03) labeled wth der by at most For the survey o graph labelg oe ca reer Mea cordal labelg was troduced by Poraj et al 4, ad the deto goes as ollows: A vertex labelg : V ( G ) {0,, } s sad to be a mea cordal labelg o G t * duces a edge labelg gve by ( u) + ( v) such that v ( ) v ( j) ad e ( ) e ( j),, j {0,, }, where v ( r) ad e ( r) deote the umber o vertces ad edges respectvely labeled wth r( r = 0,, ) A graph G s sad to be a mea cordal graph t admts a mea cordal labelg The mea cordal labelg o path, cycle, star, comb, wheel ad complete graph are dscussed 4 I ths paper, we obta some ew mea cordal graphs MEAN CORDIAL LABELING OF CATERPILLAR Deto : A caterpllar s a path, called the body, where each vertex except the ed the path may have ay umber o sgle vertces, called leaves, coected to t It s deoted bycp Fgure : Caterpllar CP Sce the caterpllars are havg ay umber o leaves ts body, t s ot easy to prove the mea cordal labelg o all types o caterpllars So we deal wth some partcular types o caterpllars ad ther mea cordal labelg We should ote that ot all caterpllars admt mea cordal labelg The star graph s a caterpllar but ot a mea cordal 4 Let us label the vertces o caterpllars as t s show Fg () Theorem : Let v V ( CP ) be a vertex the body o CP such that t () deg( vt ) = t + 3 whe = 3 t( ) () deg( vt ) = t + whe = 3t ( 7) () deg( v ) = t + whe = 3t ( 8) t I all the vertces are o degree except the ed vertces ad v t the body, the CP s mea cordal Proo: Let v, v,, v V ( CP ) Case : deg( vt ) = t + 3 Let = 3t : V ( CP ) 0,, by Dee { } ( v ) = 0, t t ( v ) =, t 3t ( vt ) = ; ( v ) =, t ad 3t 3t The v (0) = v () = v () = t ad e (0) = t, e () = e () = t Hece s a mea cordal labelg Case : deg( vt ) = t + Let = 3t : V ( CP ) 0,, by Dee { } Joural o Computer ad Mathematcal Sceces Vol 4, Issue 4, 3 August, 03 Pages (0-3)

3 Albert Wllam, et al, J Comp & Math Sc Vol4 (4), 74-8 (03) 76 ( v ) = 0, t t ( v ) =, t 3t 3 ; ( vt ) = ( v ) =, t ; ( v3 t ) = The v (0) = t, v () = v () = t ad e (0) = e () = e () = t Hece s a mea cordal labelg Case 3: deg( vt ) = t + Let = 3t : V ( CP ) 0,, by Dee { } ( v ) = 0, t t ( v ) =, t 3t ( v ) = ; ( v ) =, t t ( v ) = 3t Fgure : Mea cordal labelg o caterpllar whe = 8 The v (0) = v () = t, v () = t ad e (0) = t, e () = t, e () = t Hece s a mea cordal labelg Theorem : Let CP be a caterpllar o order ad body o legth 3 suchthat all the vertces the body have equal umber o leaves except the ed vertces The CP s mea cordal Proo: Let v, v,, v V ( CP ) Case : 0mod3 Let = 3t : V ( CP ) 0,, by Dee { } ( v ) = 0, t ; ( v ) =, t + t ( v ) =, t + 3t The v (0) = v () = v () = t ad e (0) = t, e () = e () = t Hece s a mea cordal labelg Case : mod3 Let = 3t : V ( CP ) 0,, by Dee { } ( v ) = 0, t ( v ) =, t + t ( v ) =, t 3t The v (0) = t, v () = v () = t ad e (0) = e () = e () = t Hece s a mea cordal labelg Case 3: mod3 Let = 3t Dee : ( ) { 0,,} V CP by ( v ) = 0, t ; ( v ) =, t + t ( v ) =, t + 3t Fgure 3: Mea cordal labelg o Caterpllar whe = Joural o Computer ad Mathematcal Sceces Vol 4, Issue 4, 3 August, 03 Pages (0-3)

4 77 Albert Wllam, et al, J Comp & Math Sc Vol4 (4), 74-8 (03) The v (0) = v () = t, v () = t ad e (0) = t, e () = t, e () = t Hece s a mea cordal labelg 3 MEAN CORDIAL LABELING OF SUBDIVISON OF GRAPHS Theorem 3: The graph S( P K) s a mea cordal graph Proo: Let V ( P K) = Subdvdg the edges o ( P K), we get V ( S( P K )) = 4 = m Let v, v,, v m be the vertces o V( S( P K)) Label the vertces o V ( S( P K)) as t s show Fg (4) Case : m 0mod3 : V ( S( P K )) 0,, by Dee { } ( v ) = 0, t ; ( v ) =, t + t ( v ) =, t + 3t Fgure 5: Mea cordal labelg o V ( S( P4 K)) Case : m mod3 : V ( S( P K )) 0,, by Dee { } ( v ) = 0, t ( v ) =, t + t ( v ) =, t 3t The v (0) = t, v () = v () = t ad e (0) = e () = e () = t Hece s a mea cordal labelg Case 3: m mod3 Dee : V ( S( P K )) { 0,,} by ( v ) = 0, t ; ( v ) =, t + t ( v ) =, t + 3t The v (0) = v () = t, v () = t ad e (0) = t, e () = t, e () = t Hece s a mea cordal labelg Theorem 4: The graph S( B, ) s a mea cordal graph Fgure 4: V ( S( P4 K)) The v (0) = v () = v () = t ad e (0) = t, e () = e () = t Hece s a mea cordal labelg Proo: Let V ( B, ) = + Subdvdg the edges o B,, we get V ( S( B, )) = = m Let v, v,, v m be the vertces o V ( S( B, )) Label the vertces o V ( S( B, )) as t s show Fg (6) Joural o Computer ad Mathematcal Sceces Vol 4, Issue 4, 3 August, 03 Pages (0-3)

5 Albert Wllam, et al, J Comp & Math Sc Vol4 (4), 74-8 (03) 78 Case : m 0mod3 Dee : ( ( )) { 0,,} V S B by, ( v ) = 0, t ; ( v ) =, t + t ( v ) =, t + 3t Theorem 5: The graph S( P P ), 3s mea cordal m mod3 ad m mod3 where V( S( P P )) = 5 = m Proo: Let V ( P P ) = Subdvdg the edges o ( P P ),we get V( S( P P )) = 5 = m ad E( S( P P )) = 6 4 Label the vertces o S( P P ) as t s Fg(7) Let v, v,, v m be the vertces o S( P P ) Fgure 6: GraphV ( S( B 3,3)) The v (0) = v () = v () = t ad e (0) = t, e () = e () = t Hece s a mea cordal labelg Case : m mod3 : V ( S( B )) 0,, by Dee { }, ( v ) = 0, t ( v ) =, t + t ( v ) =, t 3t The v (0) = t, v () = v () = t ad e (0) = e () = e () = t Hece s a mea cordal labelg Case 3: m mod3 : V ( S( B )) 0,, by Dee { }, ( v ) = 0, t ; ( v ) =, t + t ( v ) =, t + 3t The v (0) = v () = t, v () = t ad e (0) = t, e () = t, e () = t Hece s a mea cordal labelg Fgure 7: Graph S( P P3 ) Case : m mod3 : V ( S( P P )) 0,, by Dee { } ( v ) = 0, t ( v ) =, t + t ( v ) =, t 3t Fgure 8: Mea cordal labelg o S( P P3 ) The v (0) = t, v () = v () = t ad e (0) = t, e () = e () = t Hece s a mea cordal labelg Joural o Computer ad Mathematcal Sceces Vol 4, Issue 4, 3 August, 03 Pages (0-3)

6 79 Albert Wllam, et al, J Comp & Math Sc Vol4 (4), 74-8 (03) Case : m mod3 : V ( S( P P )) 0,, by Dee { } ( v ) = 0, t ; ( v ) =, t + t ( v ) =, t + 3t The v (0) = v () = t, v () = t ad e (0) = t, e () = e () = t + Hece s a mea cordal labelg I = = = k ad k = 6, the the total umber o vertces = 6( ) +, 6 a baaa tree s cogruet to modulo 6 Theorem 6: The graph S( P P ), 3s ot mea cordal m 0mod 3 where V ( S( P P )) = 5 = m Proo: Let V ( S( P P )) = 5 = m ad E( S( P P )) = 6 4 m Labelg t = 3 vertces o S ( P P ) wth 0, we get e (0) = t Sce E( S( P P )) = 6 4 3t, t s a cotradcto Thus, the graph s ot a mea cordal 4 MEAN CORDIAL LABELING OF BANANA TREE Deto : Let K, K,, K be a,,, k amly o stars wth the vertex sets V ( K ) = { c a a } ad deg( c ) =, k,,,, A baaa tree BT (,,, k ) s a tree obtaed by addg a ew vertex a ad jog t to a, a,, a 3 k I the ollowg theorem, we dscuss a partcular case o baaa tree where = = = k ; k = 6; V( K, ), 6 Fgure 9: Baaa tree (4, 4,4,4,4, 4) BT wth 5 vertces Theorem 7: A baaa tree BT (,,, k ) o order admts a mea cordal labelg = = = k ; k = 6 Proo: Let v, v,, v be the vertces o BT (,,, k ) Label as t s labeled Fg (9) Let = 6t + Dee : V ( BT (,,, k )) { 0,,} by ( v ) = 0, t + ( v ) =, t + 3 3t + ad 3t + 3 4t + ( v ) =, 4t + 3 5t + ad 5t + 3 6t + ( v 4t + ) = ; ( v 5t + ) = ; ( v t + ) = ; ( v + ) = 3t Joural o Computer ad Mathematcal Sceces Vol 4, Issue 4, 3 August, 03 Pages (0-3)

7 Albert Wllam, et al, J Comp & Math Sc Vol4 (4), 74-8 (03) 80 Fgure 0: Mea cordal labelg o BT (4, 4,4,4,4, 4) Fgure : Path baaa tree (4, 4,4,4,4,4) PBT wth 3 vertces The v (0) = t, v () = v () = t ad e (0) = e () = e () = t Hece s a mea cordal labelg Deto 3: Let K, K,, K be a,,, k amly o stars wth the vertex sets V( K, ) = { c, a,, a } ad deg ( c ) =, k A path baaa tree PBT (,,, k ) s a tree obtaed by addg a ew vertex a ad jog t by a subdvded edge (e a path o legth two) to a, a,, a k I the ollowg theorem, we show that a path baaa tree s admttg mea cordal labelg = = = k ; k = 6 ; V( K, ), 6 I = = = k ad k = 6, the the total umber o vertces = 6(( ) + ) +, 6 a path baaa tree s cogruet to modulo 6 Theorem 8: A path baaa tree PBT (,,, k ) o order admts a mea cordal = = = k ; k = 6 Proo Let v, v,, v be the vertces o PBT (,,, k ) Label the path baaa tree as t s labeled Fg () Let = 6t + Dee : V ( PBT (,,, k )) { 0,,} by ( v ) = 0, t + ( v ) =, t + 4 3t + ad 3t + 4 4t + 3 ( v ) =, 4t + 4 5t + ad 5t + 4 6t + ( v ) =, 5t + 5t + 3 ( v ) =, t + t + 3 ( v ) =, 3t + 3t + 3 Fgure : Mea cordal labelg o PBT (4, 4,4,4,4,4) Joural o Computer ad Mathematcal Sceces Vol 4, Issue 4, 3 August, 03 Pages (0-3)

8 8 Albert Wllam, et al, J Comp & Math Sc Vol4 (4), 74-8 (03) The v (0) = t, v () = v () = t ad e (0) = e () = e () = t Hece s a mea cordal labelg ACKNOWLEDGEMENT Ths work s supported by the Uversty Grat Commsso o Ida (UGC-MANF No: F-7/0/MANF- CHR-TAM-35) REFERENCES I Caht, Cordal graphs: A weaker verso o Graceul ad Harmoous Graphs, Ars combatoral, 3, 0-08 (987) J A Galla, A dyamc survey o graph labelg, Electroc Joural o Combatorcs, 9 (0) 3 M Hussa, E T Baskoro ad Slam, O super edge-magc total labelg o baaa trees, Utltas Math, 79, 43-5 (009) 4 R Poraj, M Svakumar, ad M Sudaram, Mea cordal labelg o graphs, Ope Joural o Dscrete Mathematcs,, (0) Joural o Computer ad Mathematcal Sceces Vol 4, Issue 4, 3 August, 03 Pages (0-3)