Adjacent Vertex Distinguishing Edge Colouring of Cactus Graphs

Transcription

1 ISSN: ISO 9:8 etfed Itetol Joul of Egeeg d Iote Techology (IJEIT) Volue 3 Issue 4 Octobe 3 Adcet Vetex Dstgushg Edge oloug of ctus Ghs Nsee Kh Mdhugl Pl Detet of Mthetcs Globl Isttute of Mgeet d Techology Kshg-74 W.B. INDIA Detet of Aled Mthetcs wth Oceology d oute Pogg Vdysg Uesty Mdoe-7 W.B. INDIA Abstct Adcet etex dstgushg edge coloug o d -coloug of gh G s g fo ts edge set to the set of oegte teges such tht o of dcet etces eet the se set of colous. The d -chotc ube deoted by ( G ) s the u ube of colous eeded d -coloug of G. A cctus gh s coected gh whch eey block s ethe edge o cycle d othe wods o edge belogs to oe th oe cycle. Hee s oed tht fo cctus gh G ( ) 3 whee G s the degee of G. A otl lgoth s lso eseted to colou the edges usg d -edge coloug techque o cctus ghs O ( ) te whee s the totl ube of etces of the cctus gh. Keywods--dcet etex dstgushg edge chotc ube dcet etex dstgushg edge coloug lyss of lgoths cctus gh desg of lgoths gh coloug I. INTRODUTION ctus gh s coected gh whch eey block s cycle o edge othe wods o edge belogs to oe th oe cycle. ctus gh he extesely studed d used s odels fo y el wold obles. Ths gh s oe of the ost useful dscete thetcl stuctue fo odellg oble sg the el wold. It hs y lctos ous felds lke coute schedulg do coucto syste etc. ctus gh he studed fo both theoetcl d lgothc ots of ew. Ths gh s subclss of l gh d sueclss of tee. Let G be sle gh wth etces. Fo d wte d fo the ube of etces of G of degee d. Let ( G) be the u ube of colous equed oe edge-coloug of G. By Vzg's theoe we kow tht ( G). A oe edge coloug s sd to be etex-dstgushg f ech of etces s cdet to dffeet set of colous. Suose tht G = ( V E ) s gh d : { } s oe edge coloug of G. Fo E c c c k y etex V let d () o sly d () deote degee of G G d ( ) = { ( w) / w E}. If u E the u s clled eghbo of d eghbo of u. We sy colou c s cdet wth etex u V ( G) f thee exsts edge ( u ) s coloued by c. A oe edge coloug s clled dcet etex dstgushg edge coloug o d -edge coloug f ( u) ( ) fo ll u E. The etex dstgushg oe edge coloug wll lso be clled s stog edge coloug. It s cle tht eey gh wthout solted edges hs d-coloug. A k - d - coloug s d-coloug usg t ost k colous. The d-chotc ube of G deoted by ( G) s the u ube of colous eeded d-coloug of G. The cocet of etex-dstgushg edge coloug hs bee cosdeed seel es [ ]. Zhg et l. [] eseted the followg coectue. oectue [] If G be sle coected gh wth t lest thee etces d G the ( G). II. REVIEW OF PREVIOUS WORK Seel esults e kow fo dcet etex dstgushg edge coloug of ghs but to the best of ou kowledge o esult s kow fo cctus gh. I ths secto the kow esults fo geel ghs d soe elted ghs of cctus ghs e eseted. A ( G ) s t lest s lge s the edge chotc ube of G t s cle tht ( G ). Blste et l. [3] oed oectue fo ll ghs wth = 3 d fo ll btte ghs. They lso showed tht the boud s tght. Oly uch weke bouds e kow fo geel ghs wthout y solted edges. Akb et l. [] obted the boud ( G ) 3 fo ll ghs wthout y solted edges. Fo ey lge Ht [] oed tht ( G ) 3 f > d Ghdeh d Ht [9] oed tht ( G ) 7 l f 6 >. Recetly Edwds et l. [8] oed tht f G s l btte gh wth ( G) the ( G ). I [3] Lu d Lu oed tht fo y coected 3-coloble Hlto gh G ( G ) 3. I [] Zhg et l. fd out the esult fo colete 6

2 ISSN: ISO 9:8 etfed Itetol Joul of Egeeg d Iote Techology (IJEIT) Volue 3 Issue 4 Octobe 3 gh K ( 3) ( K) = f (od ) d f se : If = 3k (od (od ). Fo tee T wth VT ( ) 3 ( T ) = ( T ) The we colou the edges s f y two etces of xu degee e ot dcet d f (od ( T) f T hs two etces of xu degee whch ( e ) = f (od e dcet. f (od {} f (od III. THE AVD-OLOURING OF INDUED {} f (od SUB-GRAPHS OF ATUS GRAPHS hee ( ) = {} f ( od Let G = ( V E ) be ge gh d subset U of V the duced subgh by U deoted by GU [ ] s the ge gh G= ( U E ) whee E = {( u ) : u U d ( u ) E}. Fg. : Iduce subghs of cctus gh. The cctus gh he y teested subghs those d the d-edge coloug e llustted below. A edge s deoted by P so ( edge) =. The st gh K s subgh of K theefoe oe c coclude the followg esult. Le Fo y st gh K ( K ) = whee s the degee of the st gh. III. AVD-EDGE OLOURING OF YLES A. Ad-edge coloug of oe cycle I [] Zhg et l. he colou by d-edge coloug d they he obted the followg esult. Hee we he ge costucte oe of ths esult. Le [] Fo y cycle of legth 3 f (od ( ) = 4 f (od f (od Poof: We ssue tht s cycle of legth. Let 's d e 's = be the etces d edges of esectely. To colou the edges of the cycle by d-coloug we clssfy the cycle to thee gous z. 3k 3k d 3k esectely. Hee e = ( ) e = ( ) e = ( ) fo =. Now d we colou the edges of s follows. se : If = 3k (od Hee we colou the fst 3k edges of 3k s f (od ( e ) = f (od d eg edge s f (od 3) ( e3k ) = 3. Hee the colou set of the etces fo = 3k e {} f (od {} f (od ( ) = {} f ( od {3} f = 3 k ( ) = {} f =. d se 3: If = 3k (od We colou the edges e = 3k 7 by usg the se ocess ge cse of ths le. The we coluo the eght edges by d-coloug s f = 3k 63k f = 3k 3k ( e ) = f = 3 k 43 k 3 f = 3k 33k. Ad the colou sets fo the etces = 3k 4 e se s the colou sets of the etces = 3k of the boe cse. Fo eg etces {3} f = 3k {3} f = 3k 33k ( ) = {} f = 3k {} f = 3 k. se 4: If =. The we colou the edges of s ( e ) = ( e ) = ( e ) = ( e3 ) = 3 d ( e4 ) = 4. Thus fo ll boe cses we see tht u colous eque to colou the edges of cycle e 3 whe 63

3 ISSN: ISO 9:8 etfed Itetol Joul of Egeeg d Iote Techology (IJEIT) Volue 3 Issue 4 Octobe 3 4 whe but d whe s equl to ( e ) = ( e ) = ( e ) = d ( e 3k ) = 3. esectely. se 4: If = 3k (od 3) d = 3k (od 3 f (od 4 f (od 3) d We colou the edges of s e ule ge cse Tht s ( ) = f =. of Le. Now we colou the edges e = 34 3k of s B. Ad-edge coloug of two cycles f (od Le 3 If gh G cots two cycles of fte legths ( e' ) = 3 f (od d they e oed wth coo cutetex the f (od 3) whe two cycles e of legth ( G d fo the lst edge ( e 3k ) =. ) = 4 othewse. se : If = 3k (od 3) d = 3k (od Poof: Let us cosde tht the gh G cots two cycles Hee the coloug ocess of the edges of e of legths d esectely oed by cutetex. se s ge cse of eous le. The we colou Hee degee of s 4. Ag let e = the fst 3k edges of usg se ocess of the edges d e = be the etces e = 3k ge the cse 4 of ths le. Ad d edges of d esectely we colou eg edge e 3k s ( e 3k ) =. whee e = ( ) e = ( ) d e = ( ) se 6: If = 3k (od 3) d = 3k (od = e = ( ) e = ( ) d The coloug ocess of the edges of s s se e = ( ) =. s ge cse 3 of Le. Now we colou the edges of Now we colou the edges of the gh by by usg the d-coloug ocess ge cse of ths d-coloug s follows. le. se : If = 3k (od 3) d = 3k (od Whe d both e exctly equl to the Fst we colou the edges of s e the ule of we colou the edges of the gh s d-edge coloug ge cse of eous le. Now f o = f o = we colou the edges e = 34 3k of s f o = f o = f (od f o = f o = ( e ( e ) = f (od ) = d ( e ) = 3 f o = 3 f o = 3 f (od 4 f o = 4 3 f o = 4. Ad the eg edges s f = So fo ll boe cses we see tht 4 colous e f = equed to colou the edges of the gh d colous whe ( e ) = f = legth of ech cycle s equl to. 3 f = 3k. Thus whe two cycles e of legth ( G ) = se : If = 3k (od 3) d = 3k (od 4 othewse. Hee we colou the edges of s se s ge. Ad-edge coloug of thee cycles cse of eous le. Now we colou the edges e By usg the esult of boe le we c stte the followg = 34 3k of s se ocess whch hs doe esult. the boe cse fo the edges e = 34 3k. The the Le 4 Let gh G cots thee cycles of fte legths d they e oed wth coo cutetex. If (= 6) be the degee of the ( G ) =. coloug ocedue fo the edges e = 33k e ( e ) = ( e ) = ( e ) = d ( e 3k ) = 3. se 3: If = 3k (od 3) d = 3k (od The coloug ocess of the edges of e s se s ge cse of eous le. Now we colou the edges e = 34 3k of s se ocess whch hs doe the boe cse fo the edges e = 34 3k. The we colou eg fou edges s Poof: We ssue tht the gh G cots thee cycles d esectely. They e oed by coo cutetex wth degee =6. Let e = e = d e k = be the etces edges of d esectely whee 64

4 ISSN: ISO 9:8 etfed Itetol Joul of Egeeg d Iote Techology (IJEIT) Volue 3 Issue 4 Octobe 3 e = ( ) e = ( ) d e = ( ) D. Ad-edge colog of fte ube of cycles = e = ( ) e = ( ) d Fo le 3 le 4 we c coclude the geel fo of e = ( ) =. these les whch s ge below. e = ( ) e = ( ) d e = ( ) Le If the gh G cots fte cycles of fte =. legths oed wth coo cutetex wth degee the Fst we colou two cycles ccodg to the ule ( G ) =. descbe Le 3. Now we colou the edges of s Poof: Fo Les 3 d 4 we see tht ( ) = 3 follows. ( ) = whee e two dffeet se : Fo =3k =3k d =3k =. cycles of legths oed by d Whe =3k the we colou edges e 3 ( ) = esectely whee = 3k by d 3 f (od e thee dffeet cycles of legths. Whe gh G ( e ) = f (od cots two o thee cycles of fte leghts othe th f (od oed by coo cutetex the the lue of s. So f we oe fo gh whch cots fte ube of cycles Ad the eg two edges by ( e ) = 4 d of legths the lue of s equl to degee of the ( e 3k ) =. cutetex the geel esult wll be oed utotclly. Whe = 3k the coloug schee fo the Let G cots k ube of cycles of legths edges e = 3k e (show Fg. ) oed by coo cutetex wth f (od degee. So =k. Let ( ) ( e ) = f (od k = 4 be the etces of G. Ad e e e f (od ( k ) e = 4 e the edges whee ( e ) =. 3k Ad fo the edges e d e 3k e ( e ) = 4 d Whe = 3k the we colou the edges e e 3k d e = 3k s f (od 4 f= ( e ) = d ( e ) = f (od f = 3 k f (od 3) esectely. se : Fo =3k = =3k d =3k =. Whe =3k = = 3k d = 3k the we colou the edges of ccodg to the ule of cse (fo 3k ) of ths le. Whe =3k = =3k = d = 3k the the coloug ocedue of the edges of s se s ge cse (fo 3k ) of ths le. se 3: Fo = = d =. Hee we colou the edges e = 4 s ( e ) = 3 ( e ) = ( e ) = ( e 3) = d ( e 4) =. So fo the ll boe cses we see tht sx colous (whch s equl to degee of cutetex) e eeded to colou the edges of the gh. Thus ( G ) = 6 (= ). =3. ( ) ( ) ( ) e = ( ) e4 = ( 4) d e = ( ) e = ( ) e = ( ) d ( ) ( ) ( ) ( ) 4 4 e = ( ) = k =3. Fg. : A gh G cots k ube of dffeet 's. Now we colou the edges of fst thee cycles of legths by d-coloug ccodg to the ocess s ge cse of Le 4. The we colou othe edges s follows f ( e ) = f 4 d k d f ( ) ( e ) = f f 3 d k. ( k) Hee ( e4 ) = ( k ) = k. So k colous e eeded to colou the edges of the gh G. Thus ( ) = 6

5 ISSN: ISO 9:8 etfed Itetol Joul of Egeeg d Iote Techology (IJEIT) Volue 3 Issue 4 Octobe 3 IV. AVD-EDGE OLOURING OF OTHER f (od SUBGRAPHS OF ATUS GRAPH ( e ) = f (od Le 6 Let G be gh cots fte ube of cycles f (od of fte legths d fte ube of edges oed wth coo cutetex. If be the degee of the cutetex the se : If (od 3).e. 3k. ( G ) =. The coloug ocedue of the edges of 3k e Poof: Accodg to the eous le we kow tht f se s ge cse of Le. Ad the colous of the gh cots fte ube of cycles of y legths d they edges e = 3k e ge by e oed by coo cutetex wth degee the ( G ) =. So f we oe tht f (od equl to fo G ( e ) = f (od cots cycles of legths d q ube of edges f (od oed by cutetex wth degee the we c sy tht d ( e 3k ) =. the boe stteet s tue. Hee = q. se 3: If (od 3).e. 3k Let e = q be the edges cdet o. Hee we colou the edges of 3k the coo cutetex of G. Now we colou the edges of s e the ule cycles by d-coloug usg the ule ge Le. ge cse 3 of Le. The edges e = 3k The we colou the q edges s of S s ( e ) = ( ) ( ) = q. Hee ( e q) = ( ) q = q. So ( q) colous e equed. Theefoe ( G ) =. A. Ad-edge colog of su The su gh s obted by ddg edge to ech of the etex of y cycle of fte legth. If we dd edge to ech etex of the we get su S wth etces. So s subgh of S. We oe the followg esult fo su. Le 7 Fo y su S f = ( S) = whee =3. othewse Poof: Let e = be the etces d edges of the cycle. To costuct S (show Fg. 3) we dd edge e = ( ) to the etex. The dcet etces of of e d. Ad 's e ll edet etces. To colou the edges of su by d-coloug we cosde the followg fou cses. f (od ( e ) = f (od f (od f o = 3k d ( e ) = f o = 3 k 3 k. se 4: If =. The we colou the edges e = 34 of S s f= ( e ) = f = f= 34. Thus fo ll the boe cses we fd tht 4 o colous e equed to colou the edges of su. Fo su =3. f = Theefoe ( S) = othewse. Let gh be obted fo the su S by og edge to ech of the edet etex. We obt the followg esult fo such gh. Le 8 Let G be gh obted fo S by ddg edge to ech of the edet etex the ( ) = f = G 4 othewse. Fg. 3: Su ( S ) se : If (od 3).e. 3k. Hee we colou the edges of ge cse of Le. Ad othe edges of 3k s e the ule S s The s othe ott esult of d-coloug of subgh of cctus gh whch s stted below. Le 9 Let G be gh cots cycle of y legth d fte ube of edges. If they e oed by coo cutetex wth degee the ( G ) =. oolly If the legth of the cycle s the f oe edge cdet o cutetex ( G) = f two edges cdet o cutetex. 66

6 ISSN: ISO 9:8 etfed Itetol Joul of Egeeg d Iote Techology (IJEIT) Volue 3 Issue 4 Octobe 3 ( e ) = ( e ) = ( e ) = ( e 3) = d Le If gh G cots two cycles of fte legths ( e 4) = 3 esectely. d they e oed by edge the So fo the boe cses we see tht equl to f leghts of two cycles e ( G) = whe two cycles e of legths d 4 fo othe cses. I ths othewse cse =3. hee =3. Theefoe Poof: Let d be two cycles oed by edge f leghts of two cycles e ( G ) = othewse. ( ) (show Fg. 4). Let e = e Le If gh G cots two sus d they e = be the etces d edges of d oed by oly oe edge the esectely. f leghts of two cycles e ( G) = othewse hee =3 s the degee of G. Fg. 4: The gh cots two cycles oed by edge We colou the edges of s e the ule ge Le. Ad we colou the edge ( ) by.e. ( ) =. Now the d-edge coloug ocedue of the cycle s ge bellow. se : Fo =3k =3k =. Whe =3k the we colou the fst 3k edges of s f (od ( e ) = f (od f (od d the lst edge s ( e 3k ) = 3. Whe = 3k d 3k the coloug ocess s se s boe. se : Fo = 3k =3k =. Whe = 3k the we colou the fst 3k edges of s e secod subcse of cse. We colou the eg two edges s 3 f = 3 k ( e ) = f = 3 k. Whe = 3k the we colou the fst 3k edges by usg the ule ge boe subcse. Now we colou the lst thee edges s 3 f = 3 k ( e ) = f = 3k f = 3k. se 3: Fo = 3k = 3k. Hee we colou the edges of ccodg to the ule of secod subcse of cse. Whe = d = the we colou the edges s Le Let G be gh cots oe cycle of fte legth d ech etex of the cycle cots othe cycle of fte legth. If =4 be the degee of the gh the f legth of cycle s ( G) = othewse Le 3 Fo y th P of legth f = ( P) = f = 3 3 f > 3. Poof: Let e e e be the etces d edges of P whee e = ( ) =. The we colou the edges of the th s follows: f (od f (od ( e ) = f (od So we eed d colous whe = 3 d 3 colous whe >3. f = 3 Theefoe ( P) = 3 f > 3. V. AVD-EDGE OLOURING OF ATERPILLAR Defto A ctell s tee whee ll etces of degee 3 le o th clled the bckboe of. The hlegth of ctell gh s the xu dstce of o-bckboe etex to the bckboe. The esult fo y ctell gh s ge below. Le 4 Fo y ctell gh G 67

7 ISSN: ISO 9:8 etfed Itetol Joul of Egeeg d Iote Techology (IJEIT) Volue 3 Issue 4 Octobe 3 f two etces of xu degee e ot dcet ( G) = f two etces of xu degee e dcet. Poof: Let the legth of the th of the ctell be. The we colou the edges of the th by d-coloug usg Le 3. Hee oly we oe ( G ) = whe two etces wth xu degee ( ) e dcet. The gh of Fg. s ctell. Let u be the etces wth xu degee of G d they e dcet. Tht s d( u) = d( ) =. Ag let u 's d 's be the dcet etces of u d esectely whee =. Wthout loss of geelty let the colou of the edge ( u ) be. Fg. : A ctell The ( u ) = ( u u ) = ( = d( u) ) ( ) = ( = d( ) ). So thee e.e. totl colous e equed. I the ctell ll the etces degee 3 ust le o the th. So the etces whch e t dstce 34 fo the th e of degee ethe o. Let u u u be the etces t dstce 34. So we colou the edges ( u u ) ( u u ) ( u u ) fo y of the colou such wy tht o two dcet etces he the se set of colous. The ule s sl fo the edges ( ) ( ) ( ) etc. Fo defto of lobste we kow tht lobste s oe kd of tee. So by usg the esult of d-edge coloug fo y tee [] we c coclude the esult fo lobste whch s stted below. Le Fo y lobste G f two etces of xu degee e ot dcet ( G) = f two etces of xu degee e dcet. Le 6 Let G d G be two cctus ghs. If ( G ) 3 d ( G ) 3 the ( G ) 3 whee G = G G. Poof: Let G d G be two cctus ghs d be the degees of the. Now f we ege two cctus ghs wth the etex the we get ew cctus gh G G G ). Let be the degee of G d we wll oe ( = tht x { }. Fo the gh G ( G ) 3 d fo the gh G ( G ) 3. Hee we he to oe tht the lowe d ue bouds wll esee fo the ew gh G. Let xu degee of G be tted t u d be etex of G whose degee s. Ag let u 's d 's be the dcet etces of u d esectely whee = d =. We ege G d G t d let the dcet etces of be =. If we ege oe etex of G othe th u wth of G the the dcet etces of e = x{ }. So we c sy tht les betwee x{ } d. Now fo ths esult we c coclude tht the ue d lowe bouds of e es se s G. The d-edge coloug of ll subghs of cctus ghs d the cobtos e dscussed the eous les. Fo these esults we coclude tht -lue of y cctus gh c ot be oe th 3. Hece we he the followg theoe follows. VI. AVD-EDGE OLOURING OF LOBSTER Aothe subclss of cctus gh s clled lobste gh. The defto of lobste gh s ge below. Defto A lobste s tee hg th (of xu legth) fo whch eey etex hs dstce t ost k whee k s tege. The xu dstce of the etex fo the th s clled the dete of the lobste gh. Thee e y tyes of lobstes ge ltetue lke dete dete 4 dete etc. Theoe If s the degee of cctus gh G the ( G ) 3. Poof. The d-edge coloug of ll ossble subghs of cctus gh e dscussed d he show tht ( G ) 3.. Let G be obted by -uo of two cctus ghs the G becoes cctus gh d t s oed tht ( G ) should stsfy the equlty ( G ) 3. (Le 6). Hece the theoe. 68

8 ISSN: ISO 9:8 etfed Itetol Joul of Egeeg d Iote Techology (IJEIT) VII. THE ALGORITHM AND ITS TIME OMPLEXITY To stt the lgoth of d-edge coloug of cctus gh we fst costuct gh G whch s equlet to the ge gh G. A. ostucto of equlet gh G of G Usg DFS we obt ll blocks d cutetces of cctus gh G = ( V E ). Let the blocks be B B B... BN d the cutetces be 3 R whee N s the totl ube of blocks d R s the totl ube of cutetces. The blocks of the cctus gh show Fg. 6 e { B = (7893) B = (46789) B = (33334) B = (738) B = (739) B = (7) 3 4 B = (733637) B = (9444) B = (88933) B = (6 ) B = (6 3 4 ) B = (6 6) 9 B = (4) B3 B4 = (346) = () B = () B = (67) B = (34) B = (4443)} Fg. 6: A cctus gh G d the cutetces e { } esectely. Fg. 7: The equlet gh G of G Now we he osto to costuct equlet gh G of G whose etces e the blocks of G d edge s defed betwee two blocks f they e dcet blocks of G..e. G = ( V E ) whee V = { B B BN } Volue 3 Issue 4 Octobe 3 d E = {( B B) : = = N B d B e dcet blocks }. The gh G fo the gh G of Fg. 6 s show Fg. 7. B. Ad-coloug of edges Now we tke y bty cycle s sttg block. The block s so chose tht the degee of the cutetex s xu. We deote the sttg block s B d let the leel of B be. Now the blocks dcet to B e tke t leel. The blocks dcet to the blocks of leel e tke s the blocks of leel d so o. Now we colou the block B usg the ule stted t Le. Now we colou the blocks of leel fo left to ght. Let the blocks of leel be B B B. They e ethe edges o cycles. We cosde 3 the fst block B of leel whch s dcet to B. If the block s edge the colou the block by usg Le 7. If t s cycle of fte legth the we colou t by usg Le 3. The blocks whch e dcet to B we colou the ccodg to the ule of the block B. Next we colou the edges of the block B. Suose t s ot dcet to B. The f B s edge we use Le 7 to colou the edge. If B s cycle s cycle of fte legth the Le s used. Let B be lso dcet to B. If B s edge d B s cycle the Le 6 s used d f B d B both e edges the Le 9 s used. Let us cosde the block B fo soe t leel. If t s ot dcet wth y block of leel the we colou t by Les 7 d. But f t s dcet t lest oe block of leel the we follow the ules of les 6 9 d 6. Now we colou the edges of the blocks of leel the leel 3 d so o s e the ocedue etoed boe. Suose block t leel l sy B s edge d t s dcet to block sy Bl k t leel l whch s lso edge. The we colou the block by usg the Le 8. Let block sy Bk t leel k be cycle of fte legth ts dcet block t leel k sy l B k s edge ts dcet block t leel k sy Bk s cycle of fte legth. The we colou the block Bk t leel k by usg Le. Suose block Bq s block t leel q whch s cycle of fte legth. Its dcet block sy Bq t leel q s edge. The dcet block of Bq B q t leel q s cycle of fte legth d f ts dcet block Bq s edge t leel q the we colou ths block by usg the Le. Algoth MINAVDE Iut: The cctus gh G = ( V E ). Outut: Ad-coloug of ts edges. Ste : oute the blocks d cutetces of G d 69

9 ISSN: ISO 9:8 etfed Itetol Joul of Egeeg d Iote Techology (IJEIT) Volue 3 Issue 4 Octobe 3 costuct equlet gh G of G. = O( ). Hece ste of lgoth MINAVDE tkes Ste : Let B sttg block whee the degee of the O ( ) te. cutetex of B s xu. The te colexty to colou the edges of block of sze Ste 3: We colou the block B usg Le. Ste 4: osde the blocks B = 3 of leel. olou the blocks fo left to ght s follows. () Tke the fst block B whch s dcet to B. If t s edge the we colou B by usg Le 7 d f t s cycle the we colou t by Le 3. () Next we cosde the secod block B. If t s ot dcet to B the colou t by ethe Le 7 o Le 3. If B dcet to B the colou t by usg les 6 9 d 6. () osde the block B. If t s ot dcet to y block of leel the colou t by le 7 o 3. But f t s dcet to t lest oe block of leel the follow the ules of les 6 9 d 6. () The blocks whch e dcet to B oly the colou the by the ocess sl to B. Ste : Suose block t leel l sy B l s edge d othe block Bl k t leel l dcet to B l whch s lso edge. The we colou the by usg Le 8. Ste 6: Let block Bk t leel k be cycle legth d ts dcet block t leel k be B k s edge. If ts dcet block Bk s cycle of fte legth t leel k the colou the blocks by usg Le. Ste 7: Suose Bq s block t leel q whch s cycle of fte legth. Its dcet block Bq t leel q s edge. The dcet block of Bq t leel q s B q cycle of fte legth. Let eey etex B q cots edge f Bq oe of the t leel q the we colou the blocks by usg Le. Ste 8: osde the blocks of subsequet leels d eet stes 4 to ste 7 to colou ll the etces of G ed MINAVDE. Te olexty The coectess of the lgoth follows fo the les oed the e. Theoe The te colexty of the lgoth MINAVDE s O ( ). Poof. The blocks d cutetces of y gh c be couted O( ) te [4]. Fo the cctus gh s O ( ). Ste 4 colous the etces of the blocks whch e t leel of G. If the ube of etces of ll blocks of ths leel s the the te colexty fo ste 4 s O ( ). Tht s the te colexty deeds uo the ube of etces of the whole gh. Sce the ube of etces of the ete gh s the te colexty of the lgoth s O ( ). VIII. ONLUSION The bouds of d-edge coloug of cctus gh d ous subclss z. cycle su st ctell lobste e estgted. By gg ll the esults we he obseeed tht fo y cctus gh the lue of d-edge chotc ube les betwee d 3. uetly we e egged to fd the bouds fo dffeet gh lbellg lke lst colog gceful lbelg. Hoous lbelg etc o cctus ghs. REFERENES [] M. Age E. Tesch Z Tuzu Iegul ssgets d etex-dstgushg edge-cologs of ghs otocs 9 (A. Blott et l. eds.) Elsee Scece Pub. New Yok [] S. Akb H. Bdkho d N. Nost -stog edge cologs of ghs Dscete Mth. ol [3] P. N. Blste E.Gyö J.Lehel d R.H.Schel Adcet etex dstgushg edge-cologs SIAM J. Dscete Mth. ol. No [4] P.N.Blste B.Bollbás d R.H.Schel Vetex-dstgushg cologs of ghs wth ( G) = Dscete Mth. ol. (-3) []. Bzg A.Hkt-Behde Ho L d M.Wo z k O the etex-dstgushg oe edge-cologs of ghs J. ob. Theoy B ol [6] A..Bus d R.H.Schel Vetex-dstgushg oe edge cologs J. Gh Theoy ol. 6 No [7] J. e y M. Ho ák d R. Soták Obseblty of gh Mth Sloc ol [8] K. Edwds M. Hok d M. Wozk O the eghbou-dstgushg dex of gh Gh o. ol [9] M. Ghdeh d H.Ht Two ue bouds fo the stog edge chotc ube et. [] H. Ht 3 s boud o the dcet etex dstgushg edge chotc ube J. ob. Theoy Se. B ol

10 ISSN: ISO 9:8 etfed Itetol Joul of Egeeg d Iote Techology (IJEIT) Volue 3 Issue 4 Octobe 3 [] M. Ho ák d R. Soták Obseblty of colete ulttte ghs wth quotet ts As ob ol [] M. Ho ák d R. Soták Asystotc behou of the Q obseblty of Dscete Mth ol [3] B. Lu d G. Lu O the dcet etex dstgushg edge colougs of ghs Itetol Joul of oute Mthetcs ol. 87 No [4] E. M. Regold J. Neget d N. Deo obtol Algoths : Theoy d Pctce Petce Hll Ic. Eglewood hffs New Jesy 977. [] Z.Zhg L.Lu d J.Wg Adcet stog edge colog of ghs Al. Mth. Lett. ol AUTHOR S PROFILE & Ifoto Techology Adced Modellg d Otzto Itetol Joul of Logc d outto ISRN Dscete Mthetcs d Itetol Joul of Egeeg Scece Adced outg d Bo-Techology. He s lso eewe of seel tetol ouls. Pof. Pl s the utho of the books Fot 77 wth Nuecl d Sttstcl Alyss ublshed by As Books New Delh Nuecl Alyss fo Scetsts d Egees d lsscl Mechcs ublshed by Nos New Delh d Alh Sceces Oxfod U.K. Egeeg Mthetcs Vol. I & II Adced Algeb PHI Leg New Delh Pogs cludg Nuecl d Sttstcl Methods Nos. He hs deleed ted tlks d ched tol d tetol ses/ cofeeces/ wte school/ efeshe couses Id d Abod. Addess fo coucto: Pofesso D. Mdhugl Pl Detet of Aled Mthetcs Vdysg Uesty Mdoe-7 West Begl Id El: lu@gl.co URL: htt://dysg.c./det_of_thetcs/mmp.df Moble: (+9) / D. Nsee Kh s Assstt Pofesso of Mthetcs Detet Globl Isttute of Mgeet d Techology West Begl Id. Befoe tht she ws fullte esech schol of Aled Mthetcs Detet Vdysg Uesty. She hs coleted he Phd 3 ude the gudce of Pofesso Mdhugl Pl the Pofesso of Aled Mthetcs Detet Vdysg Uesty. He B.Sc d M.Sc both wee fo Vdysg Uesty wth good ks. D. Kh hs sx tcles of whch two e tol d fou tetol ouls. He seclzto s outtol Gh Theoy. He teest fo futue esech s Fuzzy Ge Theoy Fuzzy Gh Theoy Otzto. D. Kh s the utho of the book oloug of ctus Ghs ublshed by LAP LAMBERT Acdec Publshg Gey. She s the lfelog ebe of Oeto Resech Socety (Kolkt hte). She s lso eewe of d Joul of Mthetcs. She hs tcted tol d tetol ses/ cofeeces/ wte school/ efeshe couses Id s utho d lstee. Addess fo coucto: D. Nsee Kh Aled Scece d Hutes Globl Isttute of Mgeet d Techology NH 34 Pl Moe Kshg-74 West Begl Id El: see.kh@gl.co Pof. Mdhugl Pl s cuetly Pofesso of Aled Mthetcs Vdysg Uesty Id. He hs eceed Gold d Sle edls fo Vdysg Uesty fo k fst d secod M.Sc. d B.Sc. extos esectely. Also he eceed otly wth Pof. G.P.Bhttcheee oute Dso Medl fo Isttute of Egees (Id) 996 fo best esech wok. He lso eceed Bht Jyot Awd. Pof. Pl hs successfully guded 3 esech schols fo Ph.D. degees d hs ublshed oe th tcles tetol d tol ouls. Hs seclztos clude Algothc Gh Theoy Fuzzy oelto & Regesso Fuzzy Ge Theoy Fuzzy Mtces Geetc Algoths d Pllel Algoths Fuzzy Gh Theoy. Pof. Pl s the Edto--hef of Joul of Physcl Sceces d Als of Pue d Aled Mthetcs d ebe of the edtol Bods of the ouls Itetol Joul of Fuzzy Systes & Rough Systes Itetol Joul of oute Scece Systes Egeeg 7