On Signed Product Cordial Labeling

Transcription

1 Appled Mathematcs do:.436/am..6 Publshed Ole December ( O Sed Product Cordal Label Abstract Jayapal Baskar Babujee Shobaa Loaatha Departmet o Mathematcs Aa Uversty Chea Ida E-mal: {baskarbabujee shobaa_}@yahoo.com Receved September 9 ; revsed October 5 ; accepted November 3 A ew cocept o label called the sed product cordal label s troduced ad vestated or path raph cycle raphs star-k Bstar-B P 3 ad C 3. Some eeral results o sed product cordal label are studed. Keywords: Graph Label Fucto Cordal Label. Itroducto I the vertces o the raph are assed values subject to certa codtos the t s kow as raph label. Most o the raph label problems have the ollow three commo characterstcs: a set o umbers or assmet o vertex labels a rule that asss a label to each ede ad some codto(s) that these labels must satsy. Cordal labels were troduced by Caht [] who called a raph G cordal there s a vertex label : V( G) such that the duced label : E( G) deed by ( xy) ( x) ( y) or all edes xy E( G) ad wth the ollow equaltes hold: v () v () ad e () e () where v () (respectvely e () ) s the umber o vertces (respectvely edes) labeled wth. Sudaram ad Somasudaram [] troduced the oto o product cordal labels. A product cordal label o a raph G wth vertex set V s a ucto rom V to such that each ede uv s assed the label ( u) ( v ) the umber o vertces labeled wth ad the umber o vertces labeled wth der by at most ad the umber o edes labeled wth ad the umber o edes labeled wth der by at most. A raph wth a product cordal label s called a product cordal raph. For detal survey o raph label oe ca reer Galla [3]. Let G ( V E) be a raph. As ve [4] a mapp : V( G) s called bary vertex label o G ad () v s called the label o the vertex v o G uder. For a ede e uv the duced ede label- : E( G) s ve by () e () u (). v Let v () v () be the umber o vertces o G hav labels ad respectvely uder ad let e () e () be the umber o edes hav labels ad respectvely uder. A bary vertex label o a raph G s called a cordal label v () v () ad e () e ( ). I Secto we troduce the deto o sed product cordal label ad work or ew udametal raphs. I Secto 3 we prove that P C ad bstar raph B are sed product cordal. Fally Secto 4 we prove the exstece o sed product cordal label or some eeral raphs.. Sed Product Cordal Label We ow troduce the deto o sed product cordal label. Deto.: A vertex label o raph G : V( G) { } : E( G) { } wth duced ede label deed by ( uv) ( u) ( v) s called a sed product cordal label v ( ) v () ad e ( ) e () where v ( ) s the umber o vertces labeled wth v () s the umber o vertces labeled wth e ( ) s the umber o edes labeled wth ad e () s the umber o edes labeled wth. A raph G s sed product cordal t admts sed product cordal label. Theorem.: The Path raph P admts sed product cordal label. Proo: Let V v v... v be the vertex set ad E vv ; be the ede set o the path raph Copyrht ScRes.

2 56 J. B. BABUJEE ET AL. P. To dee vertex label : V( G) { } the ollow cases are to be cosdered. Case : whe 3 (mod 4) or ; mod4 ( v ) ; 3 mod 4 Case : whe (mod 4) or ad ; mod4 ( v ) ; 3 mod 4 ( v ) ( v ) The duced ede label vv v v ( ) ( ) ( ) s ve by ( v) ad ( v ) have same s ( v ) ad ( v ) have deret s wth respect to the above label patter we ve the proo as ollows ) Whe mod4 The total umber o vertces labeled wth s are ve by v ( ) ad the total umber o vertces labeled wth s are ve by v (). Thereore the total derece betwee the vertces labeled wth s ad s s v ( ) v (). The total umber o edes labeled wth s are ve by e ( ) ad the total umber o edes labeled wth s are ve by e (). Thereore the total derece betwee the edes labeled wth s ad s s e ( ) e () der by oe. v ( ) v () e ( ) e () ) Whe mod 4 The total umber o vertces labeled wth s are ve by v ( ) ad the total umber o vertces labeled wth s are ve by v (). Thereore the total derece betwee the vertces labeled wth s ad s s v ( ) v (). The total umber o edes labeled wth s are ve by e ( ) ad the total umber o edes labeled wth s are ve by e (). Thereore the total derece betwee the edes labeled wth s ad s s e ( ) e () der by oe. ) Whe s odd v ( ) v () e ( ) e () The total umber o vertces labeled wth s are ve by v ( ) ad the total umber o vertces labeled wth s are ve by v (). Thereore the total derece betwee the vertces labeled wth s ad s s v ( ) v (). The total umber o edes labeled wth s are ve by e ( ) ad the total umber o edes labeled wth s are ve by e (). Thereore the total derece betwee the edes labeled wth s ad s s e ( ) e () der by zero. v ( ) v () e ( ) e () Thus each cases we have v ( ) v () ad e ( ) e (). Hece the path raph P ad- mts sed product cordal label. Theorem.3: The Cycle raph C 3 admts sed product cordal label except whe mod4. Proo: Let V v v v be the vertex set ad E vv ; vv be the ede set o the cycle raph C. To dee vertex label : V( G) { } the ollow case s to be cosdered. Whe 3 (mod 4) or ; mod4 ( v ) ; 3 mod 4 The ede label s ve by ( vv ) ( v) ( v ) ( v) ad ( v ) have same s ( v) ad ( v ) have deret s I vew o the above label patter we have Table. Table. Vertex ad ede codtos o a cycle raph. v ( ) v () v ( ) v () mod4 mod4 3mod4 e ( ) () e e e ( ) () mod4 mod4 3mod4 Copyrht ScRes.

3 J. B. BABUJEE ET AL. 57 Hece the cycle raph C; mod 4admts sed product cordal label. Theorem.4: The star raph K admts sed product cordal label. Proo: The star raph K s a tree obtaed by add pedet ede to the ceter vertex. Let V v v v v be the vertex set ad the ede set s ve by E v v ;. To dee vertex label or : V( G) { } s ve by or ; mod ( v ) ; mod Whe s eve ad odd the ede label s ve by ( vv ) ( vv ) ; ad ( vv ) ; vv ( ) ; respectvely. I vew o the above label patter we have Table. Hece the star raph K admts sed product cordal label. 3. Sed Product Cordal Label or Specal Graphs I ths secto we prove the sed product cordal label or the raphs P C ad the bstar raph B. Theorem 3.: The Path raph P 3 admts sed product cordal label. Proo: Let v v v3 v ad u u u 3 u be the vertex sets o the path raph P ad the ede set s ve by E vv ; E vu ; The raph P has vertces ad edes. To dee vertex label : V( G) { } the ollow Table. Vertex ad ede codtos o a star raph. v ( ) v () v ( ) v () mod mod e ( ) () e e e ( ) () mod mod 3 cases are to be cosdered. Case : whe s eve or ; mod4 ( v ) ; 3 mod 4 ; mod ( u ) ; mod The ede label are deed as ollows or ( vv ) ( v) ( v ) ( v) ad ( v ) have same s ( v) ad ( v ) have deret s or ( vu ) ( v) ( u) ( v) ad ( v ) have same s ( v) ad ( v ) have deret s Case : Whe s odd The vertex label s ve by or or ; mod 4 ( v ) ; 3 mod 4 ( v ) ; ( v ) ; mod ( u ) ; mod ( u ) The ede label are deed as ollows or vv v v ( ) ( ) ( ) ( v) ad ( v ) have same s ( v) ad ( v ) have deret s or vu v u ( ) ( ) ( ) ( v) ad ( v ) have same s ( v) ad ( v ) have deret s I vew o the above label patter we have Table 3. Hece the raph P 3 admts sed product cordal label. Theorem 3.: The raph C 3 admts sed product cordal label except or mod 4. Copyrht ScRes.

4 58 J. B. BABUJEE ET AL. Table 3. Vertex ad ede codtos o the raph P 3. v ( ) v () v ( ) v () mod mod4 3mod4 e ( ) () e e e mod mod4 3mod4 ( ) () Proo: Let v v v3 v ad u u u 3 u be the vertex sets o the raph C 3 wth vertces ad edes. The ede set s ve by E vv ; vv E vu ; The vertex label s deed by or ; mod4 ( v ) ; 3 mod 4 ; mod ( u ) ; mod The ede label s ve by or ( vv ) ( v) ( v ) ( v) ad ( v ) have same s ( v ) ad ( v ) have deret s v v ( ) ad ( ) have same s ( vv ) ( v ) ad ( v ) have deret s ( v) ad ( v ) have same s ( vu ) ( v) ad ( v ) have deret s I vew o the above label patter we have the total umber o vertces labeled wth s are ve by v ( ) ad the total umber o vertces labeled wth s are ve by v (). Thereore the total derece betwee the vertces labeled wth s ad s s v ( ) v (). The total umber o edes labeled wth s are ve by e ( ) ad the total umber o edes labeled wth s are ve by e (). Thereore the total derece betwee the edes labeled wth s ad s s e ( ) e () der by zero. v ( ) v () e ( ) e () Hece the cycle raph C 3 admts sed product cordal label. Theorem 3.3: The raph B admts sed product cordal label. Proo: The raph B s a bstar obtaed rom two dsjot copes o K by jo the cetre vertces by a ede. It has + vertces ad + edes. Let v v v 3 v be the vertces o the bstar raph. The ede set s deed as E vv ; E v v ; We ow dee vertex label : V( G) { } as ; mod ; ( v ) ; mod ; 3 ( v ) ( v ) The ede label s ve by ( vv ) ; ( vv ) ( vv ) ; The total umber o vertces labeled wth s are ve by v ( ) ad the total umber o vertces labeled wth s are ve by v (). Thereore the total derece betwee the vertces labeled wth s ad s s v ( ) v (). The total umber o edes labeled wth s are ve by e ( ) ad the total umber o edes labeled wth s are ve by e (). Thereore the total derece betwee the edes labeled wth s ad s s e ( ) e () der by oe. I vew o the above label patter we have v ( ) v () e ( ) e () From the above label patter we have v ( ) v () ad e ( ) e (). Hece the bstar raph B admts sed product cordal label. Example 3.4: Fure llustrates the sed product cordal label or Bstar B55. Amo the eleve edes ve edes receve the label + ad sx edes receve the label. Copyrht ScRes.

5 J. B. BABUJEE ET AL. 59 Fure. Sed product cordal label o Bstar B Geeral Results o Sed Product Cordal Label I ths secto we prove the Sed product cordal label or the some eeral raphs. Deto 4.: The tree T@mK s obtaed by attach m copes o K to ay oe o the vertces T. Theorem 4.: I a tree T admts sed product cordal label wth vertces the T@mK (m eve) s also a sed product cordal tree. Proo: Let us assume that a tree T admts sed product cordal label. The mapp : V { } satses the codto o sed product cordal label. Let TT@ mk ( V E) where V V u u u ad E E vu vu vu. The vertex label or T : V { } s deed as () v () v vv. Let u u um be the m vertces joed to the vertex v o the tree T. The vertex label o u u um s ve by ( u ) ( ) ; m. I () v the the s o ( vu) ( vu) ( vu m ) wll be the s o ( u) ( u) ( um ) respectvely ad (v) = the the s o ( vu) ( vu) ( vu m ) wll be the s o ( u) ( u) ( um ). I both the cases the ew m edes vu vu vum cotrbutes equal umber o edes labeled wth s ad s. Thereore a tree T@mK the derece betwee the total umber o vertces labeled wth s ad s ad the derece betwee the total umber o edes labeled wth s ad s ders by utmost oe. Example 4.3: Fure llustrates the sed product cordal label or T@4K where T s a arbtrary tree hav sed product cordal label. Corollary 4.4: I a coected raph G has sed product cordal label the the raph G@mK where m s eve admts sed product cordal label. Deto 4.5: The tree Tˆ P s obtaed by supermpos a pedat vertex o P wth ay o the selected vertex T. Theorem 4.6: I a tree T as sed product cordal label the Tˆ P where m s odd admts sed product cordal label. Fure. Sed product cordal label o T@4K. Proo: Let T ( V E) be a coected raph wth vertex set V ad ede set E the the tree Tˆ P ( V E ) where V V u u u ad E Eu u ;. Let v be a vertex T. we supermpose a pedat vertex wth a selected vertex v T oly they preserves the same s. Case : I () v ( u ) the the vertex label or the path raph ollows rom theorem. Case : I () v ( u ) the the vertex label s ve by Sub Case.: whe 3 (mod 4) or Sub Case.: whe or ad ( v ) ; 3mod4 ; mod 4 (mod4) ( v ) ; 3mod4 ; mod 4 ( v ) ( ) v As we supermpose the pedat vertex o P wth ay oe o the vertex T provded they preserve the same s the the path raph cotrbutes equal umber o vertces ad edes labeled wth s ad s. Hece the tree Tˆ P admts sed product cordal label. Corollary 4.7: I a coected raph G has sed product cordal label the the raph Tˆ P where s odd admts sed product cordal label. Theorem 4.8: I a coected raph G has a sed product cordal label wth vertces ( mod 4) the G + admts sed product cordal. Proo: Let G ( V E) be a coected raph wth vertex set V u u u. Sce G has a sed product cordal label there exsts : V{ } such that v ( ) v () ad e ( ) e (). Copyrht ScRes.

6 53 J. B. BABUJEE ET AL. As mod4 v ( ) v (). Let V VV where V u : mod ad V u : mod. For our coveece let us assume that maps all the vertces o V to ad all the vertces o V to. The raph G V E s obtaed by attach the pedat vertces u u u to each o the vertces u u u G. Let the vertex set ad ede set o G be deed as Vu u : ad E E uu ;. The vertex label or the raph G : V { } s deed as ollows or u ( ) ( u ) ( u ) whe modad ( u ) mod4 whe mod ad ( u ) mod4 whe modad ( u ) mod4 whe mod ad ( u ) 3mod4 The duced ede labels ( uv) ( uv) uv E. All the ewly added edes ( uu ) wll share equally the labels ad as per our costructo above. Hece v ( ) v () ad e ( ) e (). Oly by ths label patter the raph G admts sed product label where s a multple o Ackowledemets The reeree s rateully ackowleded or ther suestos that mproved the mauscrpt. 6. Reereces [] I. Caht Cordal Graphs: A Weaker Verso o Graceul ad Harmoous Graphs Ars Combatora Vol pp. -7. [] M. Sudaram R. Poraj ad S. Somasudram Total Product Cordal Label o Graphs Bullet o Pure ad Appled Sceces: Secto E. Mathematcs ad Statstcs Vol. 5 6 pp [3] J. A. Galla A Dyamc Survey o Graph Label Electroc Joural o Combatorcs Vol. 7 No. DS6 pp [4] S. K. Vadya N. A. Da K. K. Kaa ad P. L. Vhol Cordal ad 3-Equtable Label or Some Star Related Graphs Iteratoal Mathematcal Forum Vol. 4 No. 3 9 pp Copyrht ScRes.