Itratoal Joural Of Computatoal Egrg Rsarch (crol.com) Vol. Issu. 5 Total Prm Graph M.Rav (a) Ramasubramaa 1, R.Kala 1 Dpt.of Mathmatcs, Sr Shakth Isttut of Egrg & Tchology, Combator 641 06. Dpt. of Mathmatcs, Maomaam Sudaraar Uvrsty, Trulvl 67 01. Abstract: W troduc a w typ of lablg kow as Total Prm Lablg. Graphs whch admt a Total Prm lablg ar P K calld Total Prm Graph. Proprts of ths lablg ar studd ad w hav provd that Paths, Star 1,, Bstar, C Comb, Cycls H K whr s v, Hlm,,, t C ad Fa graph ar Total Prm Graph. W also prov that ay cycl C whr s odd s ot a Total Prm Graph. Kywords: Prm Lablg, Vrtx prm lablg, Total Prm Lablg, Total Prm Graph 1. Itroducto By a graph G = (V,E) w ma a ft, smpl ad udrctd graph. I a Graph G, V(G) dots th vrtx st ad E(G) dots th dg st. Th ordr ad sz of G ar dotd by p ad q rspctvly. Th trmology followd ths papr s accordg to [1]. A lablg of a graph s a map that carrs graph lmts to umbrs. A complt survy of graph lablg s []. Prm lablg ad vrtx prm lablg ar troducd [4] ad [6]. Combg ths two, w df a total prm lablg. Two tgrs a ad b ar sad to b rlatvly prm f thr gratst commo dvsor s 1, (..) ab, 1. Ifa, a 1, for all 1, th th umbrs a1, a, a,, a ar sad to b rlatvly prm pars. Rlatvly prm umbrs play a vtal rol both aalytc ad algbrac umbr thory. Dfto 1.1 [4] Lt G=(V,E) b a graph wth p vrtcs. A bcto f : V( G) 1,,,, p s sad to b as Prm Lablg f for ach dg =xy th labls assgd to x ad y ar rlatvly prm. A graph whch admts prm lablg s calld Prm Graph. Dfto 1. [6] Lt G=(V,E) b a graph wth p vrtcs ad q dgs. A bcto f : E( G) 1,,,, q s sad to b a Vrtx Prm Lablg, f for ach vrtx of dgr at last two, th gratst commo dvsor of th labls o ts cdt dgs s 1.. Total Prm Graph: Dfto.1 Lt G=(V,E) b a graph wth p vrtcs ad q dgs. A bcto f : V E 1,,,, p q sad to b a Total Prm Lablg f () for ach dg =uv, th labls assgd to u ad v ar rlatvly prm. () for ach vrtx of dgr at last, th gratst commo dvsor of th labls o th cdt dgs s 1. A graph whch admts Total Prm Lablg s calld Total Prm Graph. Exampl. (1) C 4 s a Total Prm Graph. 5 1 8 6 s 4 7 Iss 50-005(ol) Sptmbr 01 Pag 1588
Itratoal Joural Of Computatoal Egrg Rsarch (crol.com) Vol. Issu. 5 () C (or) K s ot a Total Prm Graph, bcaus w ca assg oly o v labl to a dg ad o mor v labl to a vrtx. But w hav totally thr v labls ad th thrd v labl ca b assgd thr to ay vrtx ot to ay dg. Not that C has Prm Lablg as wll as Vrtx Prm Lablg. Notatos. (1) ad dots th maxmum ad mmum dgr of a vrtx rspctvly. () () dots th gratst tgr lss tha or qual to. dots th last tgr gratr tha or qual to. (4) g.c.d dots gratst commo dvsor. Thorm.4 Th path P v v v v Proof Lt 1 P s a Total Prm Graph. P has vrtcs ad -1 dgs.. W df f : V E 1,,,,( 1) Clarly f s a bcto. f v,1 f,1 Accordg to ths pattr, th vrtcs ar labld such that for ay dg =uvg, gcd [ f u, f v] 1. Also th dgs ar labld such that for ay vrtx v, th g.c.d of all th dgs cdt wth v s 1. Hc P s a Total Prm graph. Dfto.5 K 1 wth pdt dgs cdt wth 1 Thorm.6 1,, 1 Proof Lt V( K ) u K s a Total Prm Graph. 1 ad v,1 b th vrtcs adact to u. Thrfor K 1, has +1 vrtcs ad dgs. Now w df f : V E 1,,,,( 1) f u Clarly f s a bcto. V( K ) s calld a Star Graph ad s dotd by 1, 1 f v,1 f 1,1 Accordg to ths pattr, K1, s a Total Prm Graph. Dfto.7 Th graph obtad from K 1, ad K1,m by og thr ctrs wth a dg s calld a Bstar ad s dotd by B(,m) Thorm.8 Bstar B(,m) s a Total Prm Graph. V( K ) u, v ad u,1 ; v,1 m b th vrtcs adact to u ad v rspctvly. Proof Lt For 1, uu uv ; for ( ) ( m 1), vv ; 1 Thrfor B(,m) has +m+ vrtcs ad +m+1 dgs. Now w df f : V E 1,,,,( m ). K. Iss 50-005(ol) Sptmbr 01 Pag 1589
Itratoal Joural Of Computatoal Egrg Rsarch (crol.com) Vol. Issu. 5 1 1,1 f u f v f u 1,1 f v m f m,1 ( m 1) Clarly f s a bcto. Accordg to ths pattr, clarly B(,m) s a Total Prm Graph. Dfto.9 A graph obtad by attachg a sgl pdt dg to ach vrtx of a path 1 Comb. Thorm.10 Comb s a Total Prm Graph. Proof Lt G b a Comb obtad from th path by og a vrtx u to v, 1. Th dgs ar labld as follows: For1, 1 vu vv 1 Thrfor G has vrtcs ad -1 dgs. Now df f : V E 1,,,,(4 1) f v 1,1 f u,1 f,1 ( 1) Clarly f s a bcto. Accordg to ths pattr, Comb s a Total Prm Graph. Thorm.11 Cycl C, s v, s a Total Prm Graph. Proof Lt C v v v v v 1 1 1 Thrfor C has vrtcs ad dgs. Now w df f : V E 1,,,, Clarly f s a bcto. f v,1 f,1 Accordg to ths pattr, clarly Cycl C, s v, s a Total Prm Graph. Thorm.1 Cycl C, s odd, s ot a Total Prm Graph. Proof Lt C v v v v v 1 1 1 Thrfor C has vrtcs ad dgs. Df f : V E 1,,,, P v v v v s calld a Now, o. of v labls avalabl s. For ay coscutv vrtcs, w ca assg at most o v labl ad so, umbr of vrtcs wth v labls s at most. Iss 50-005(ol) Sptmbr 01 Pag 1590
Itratoal Joural Of Computatoal Egrg Rsarch (crol.com) Vol. Issu. 5 Also, out of coscutv dgs, w ca assg at most o v labl ad so th umbr of dgs wth v labls s at most. Thrfor th cssary codto for xstc of total Prm Graph s =. Cas 1: 0mod (..) s a multpl of. Thrfor, ths cas (..) (..) Whch s a cotradcto. Cas : 1mod I ths cas (..) 4 (..) 4 But s odd, so t s ot possbl. mod Cas : 1 I ths cas (..) (..) But s odd, so t s ot possbl. Thrfor Cycl C, s odd, s ot a Total Prm Graph. Dfto.1 Hlm H s a graph obtad from whl by attachg a pdt dg at ach vrtx of -cycl. Thorm.14 Hlm H s a Total Prm Graph. u, u,, u Proof Hr ctr vrtx wll b labld as u ad all th vrtcs o th cycl ar labld as 1. Th corrspodg pdt vrtcs ar labld as v 1, v,, v. Th dgs ar labld as 1,,, startg from th pdt dg cdt at vrtx u 1 ad wth lablg th dg o th cycl altratvly clockws drcto 1,,, ad th spoks of th whls ar labld as 1,,, startg from th dg uu1 ad procdg th clockws drcto. Thrfor Hlm H has +1 vrtcs ad dgs. Now w df f : V E 1,,,,(5 1) f u 1 1 f u 1,1 1 f v,1 f 1,1 Iss 50-005(ol) Sptmbr 01 Pag 1591
Itratoal Joural Of Computatoal Egrg Rsarch (crol.com) Vol. Issu. 5 Clarly f s a bcto. Accordg to ths pattr, clarly Hlm H s a Total Prm Graph. Dfto.15 K m, s a complt bpartt graph wth bpartto X ad Y, whch ay two vrtcs X as wll as ay two vrtcs Y ar o-adact. Also vry vrtx of X s adact to vry vrtx of Y. Thorm.16 K,, s a Total Prm Graph. Proof K m, hav m+ vrtcs ad m dgs. Hr m=. Thrfor K, has + vrtcs ad dgs. Lt X u, u ad Y v v v v v u 1 to 1 1 ad th last dg 1 1,,,,. Th dgs ar labld a cotuous mar startg from 1 v 1 u 1 v u. Now w df f : V E 1,,,,( ) as follows: f u 1,1 Th vrtcs Y v v v v 1,,,, ar parttod to sts as follows: for v ad 0 S v 1, v f v Th dgs ar labld as follows:- Cas 1: s odd () for 0 k 1,1, lt f v ad v S +1, s odd +, s v +1, s of th form 10r- ad r=1,,, u v, 4k1 1 4k1 u v 4k 1 k u v 4k 4k () for 0 k, 4k4 uvk u v, u v, u v () 1 1 1 Cas : s v () for 0 k 1, 4k 1 u1v 4k1 u v 4k 1 k u v 4k 4k () for 0 k, 4k4 uvk () uv1 Iss 50-005(ol) Sptmbr 01 Pag 159
Itratoal Joural Of Computatoal Egrg Rsarch (crol.com) Vol. Issu. 5 Th uassgd labls gv thr ordr to th dgs th ordr Clarly f s a bcto. 1,,,,. Accordg to ths pattr, clarly K,, s a Total Prm Graph. t t Dfto.17C dots th o-pot uo of t cycls of lgth. C s also calld as Frdshp Graph (or) Dutch t-wdmll. t Thorm.18 s a Total Prm Graph. C t Proof C has t+1 vrtcs ad t dgs. Lt th vrtx st bv v v v wth ctr vrtx v 0. Lt th dg st b,,,, t 0 1 labl th dgs clockws drcto. Now w df f : V E 1,,,,(5t 1) Clarly f s a bcto. Accordg to ths pattr, clarly C as follows: f v 1,0 t 1 f t 1,1 t t s a Total Prm Graph. Dfto.19 Th fa graph F s dfd as K1 P, Thorm.0 Fa graph F,, s a Total Prm Graph. Proof F has 1 vrtcs ad -1 dgs. f : V E 1,,,, W df v v, 1 For 1 For v1v, 1 Clarly f s a bcto. P s a path of vrtcs.,1 f v 1 f f Accordg to ths pattr, Fa Graph F s a Total Prm Graph. 1,,,, t wth 1 v0v1 ad Rfrcs [1]. F.Harary, Graph Thory, Addso Wsly, Radg, Mass., 197. []. J.A.Galla, A dyamc survy of graph lablg, Elctroc J.Combatorcs, (Jauary 010). []. T.M.Apostol, Itroducto to Aalytc Numbr Thory, Narosa Publshg Hous, 1998. [4]. A.Tout, A.N. Dabbouy ad K.Howalla, Prm Lablg of graphs, Natoal Acadmy Scc Lttrs, 11(198), 65-68. [5]. Fu,H.L ad Huag,K.C (1994) o Prm Lablg Dscrt mathmatcs, North Hollad, 17, 181-186.. [6]. T.Drtsky, S.M.L ad J.Mtchm, o Vrtx prm lablg of graphs graph thory, Combatorcs ad Applcatos Vol.1, J.Alav, G.Chartrad ad O.Ollrma ad A.Schwk, ds, Procdgs 6 th tratoal cofrc Thory ad Applcato of Graphs (Wly, Nwyork,1991) 59-69. Iss 50-005(ol) Sptmbr 01 Pag 159