Hamiltonian-T*- Laceability in Jump Graphs Of Diameter Two

Transcription

1 IOSR Jourl of Mthemtics IOSR-JM e-issn p-issn9-76x. Volume Issue Ver. III My-Ju. PP -6 Hmiltoi-T*- Lcebility i Jump Grphs Of Dimeter Two Mjuth.G Murli.R Deprtmet of MthemticsGopl college of Egieerig d Mgemet Bglore. Deprtmet of Mthemtics Dr.Ambedkr Istitute of Techology Bglore Abstrct Let G be the oempty grph. The Jump grph J[G] of G is the grph whose vertices re edges of G d where two vertices of J[G] re djecet if d oly if they re ot djcet i G equivletly the Jump grph J[G] is the complémet of lie grph L[G]. I [] the uthors hve obtied Hmiltoi Jump grphs. I this pper we chrecterised Hmiltoi lcebility of Jump grph d lso explore the Lcebility i jump grph of Str Friedship d Ldder grphs. Keywords Coected Grph Jump grph Friedship grph Ldder grphhmiltoi-t*-lceble Grph. I. Itroductio Let G be fiite simple coected d udirected grph. Let u d v be two vertices i G. The distce betwee u d v deoted by duv is the legth of shortest u-v pth i G. G is Hmiltoi lceble if there exists Hmiltoi pth betwee every pir of vertices i G t odd distce. G is Hmiltoi-tlceble Hmiltoi-t*-lceble if there exists Hmiltoi pth betwee every pir t lest oe pir of vertices u d v i G with the property duv=t t dimg. The Lie grph L[G] of G hs the edges of G s its vertices d two vertices of L[G] re djcet if d oly if they re djcet i G. The complemet of the lie grph L[G] is clled the Jump grph J[G] of G. Tht is the Jump grph J[G] is the grph defied o E[G] d i which two vertices re djcet if d oly if they re ot djcet i G. I [] [] d [] the uthors hve studied Hmiltoi-t-lcebility d Hmiltoi-t*-lcebility of vrious grph structures. I this pper we explore the Hmiltoi-t*-lceblity properties of the Jump grph J[G] of the Str grph the Friedship grph d the Ldder grphs. Defiitio The Jump grph J [G] of grph G is grph defied o E [G] d i which two vertices is djcet if d oly if they re ot djcet i G. Defiitio A Grph is clled Friedship grph if every pir of its odes hs exctly oe commo eighbor. This coditio is clled the friedship coditio. Furthermore grph is clled widmill grph if it cosists of k trigles which hve uique commo ode kow s the politici. Clerly y widmill grph is friedship grph The Fried ship Grph or Dutch Widmill grph or -F grph F is plr udirected grph with + vertices d edges. Defiitio The Ldder grph L is plr udirected grph with vertices d +- edges. Figures d below illustrte the Jump Grph of the Str K the Friedship grph F d the Ldder Grph L respectively. Fig. K L[K ] J[K ] Pge

2 Hmiltoi-T*- Lcebility i Jump Grphs Of Dimeter Two Fig. F L [F ] J [F ] Fig. L L[L ] J[L ] II. Results Theorem The Jump Grph J[G] of the str Grph G =K us Hmiltoi-t*-lceble for t dimg; 6. Proof Cosider Jump Grph J[G] which is complemet of Lie grph of G deote the vertices of J[G] by uder modulo. Now we hve the followig cses Cse For t= Sub cse If is eve I J[G] P d d the pth is Hmiltoi pth from to. Fig. Hmiltoi pth from the vertex to i Jump Grph J [K 8 ] Sub cse If is odd Pge

3 Hmiltoi-T*- Lcebility i Jump Grphs Of Dimeter Two 7 Pge I J[G] d d the pth 6 P 7 6 is Hmiltoi pth. Sub cse If is odd =7 I J[G] d d the pth 6 6 P is Hmiltoi pth Fig. Hmiltoi Pth from the vertex to i Jump Grph J [K 7 ] Cse For t==dimj[g] Sub cse If is eve I J[G] d d the pth 6 P is Hmiltoi pth from to. Fig.6 Hmiltoi Pth from the vertex to i Jump Grph J [K ] Sub cse If is odd I J[G] d d the pth 7 P 6 is Hmiltoi pth from to. Therefore G is Hmiltoi-t*-lceble for t=.

4 Hmiltoi-T*- Lcebility i Jump Grphs Of Dimeter Two Fig.7 Hmiltoi Pth from the vertex to i Jump Grph J [K ] Remrks. If J[G] is either trivil or discoected grph.. If = the i J[G] we cosider the followig cses For t= d i d the the pth P is Hmiltoi pth. Fig.8. Hmiltoi pth from the vertex to i Jump Grph J [K ] For t= Hmiltoi Pth does ot exists. Theorem The Jump Grph J[G] where G=F the Friedship Grph is Hmiltoi-t*-lceble for t dimg. Proof Cosider Jump Grph J [G] which is complemet of Lie grph of G deote the vertices of J[G] by. Cse For t= Sub cse If is eve I J [G] d d the pth P is Hmiltoi pth. Hece J [G] is Hmiltoi-t*-lceble for t=. 8 Pge

5 Hmiltoi-T*- Lcebility i Jump Grphs Of Dimeter Two Fig.8. Hmiltoi pth from the vertex to i Jump Grph J [F ] Sub cse If is odd I J [G] d d the pth P lceble for t= is Hmiltoi pth. Hece J[F ] is Hmiltoi- t*- 7 7 Cse For t= Sub cse If is eve I J [G] Fig.9. Hmiltoi pth from the vertex to i Jump Grph J [F ] is Hmiltoi pth. Hece J[F ] is Hmiltoi- t*-lceble for t= d d the pth P 9 Pge

6 Hmiltoi-T*- Lcebility i Jump Grphs Of Dimeter Two Fig. Hmiltoi pth from the vertex to i Jump Grph J [F ] Cse ii If is odd I J [G] d d the pth P for t=. is Hmiltoi pth. Hece J[F ] is Hmiltoi- t*-lceble Fig. Hmiltoi pth from the vertex to i Jump Grph J [F ] Theorem The Jump Grph J[G] where G=L the Ldder grph is Hmiltoi-t*-lceble for t dimg. Proof Cosider Jump Grph J [G] which is complemet of Lie grph of G deote the vertices of J[G] by. Now we hve the followig cses Cse For t= Sub cse If is eve 6 Pge

7 I J [G] Hmiltoi-T*- Lcebility i Jump Grphs Of Dimeter Two d d the pth P is Hmiltoi pth. Hece J[L ] is Hmiltoi- t*-lceble for t=. Fig. Hmiltoi pth from the vertex to i Jump Grph J [L ] Sub cse If is odd I J [G] P 6 8 d d the pth Hece J[L ] is Hmiltoi- t*-lceble for t= is Hmiltoi pth. Fig. Hmiltoi pth from the vertex to i Jump Grph J [L ] Cse For t= Sub cse If is eve 6 Pge

8 P Hmiltoi-T*- Lcebility i Jump Grphs Of Dimeter Two I J [G] d d the pth for t= is Hmiltoi pth. Hece J[L ] is Hmiltoi- t*-lceble Fig. Hmiltoi pth from the vertex to i Jump Grph J [L 6 ] Sub cse If is odd I J [G] d d the pth P pth. Hece J [L ] is Hmiltoi- t*-lceble for t= is Hmiltoi Fig. Hmiltoi pth from the vertex to i Jump Grph J [L ] 6 Pge

9 Hmiltoi-T*- Lcebility i Jump Grphs Of Dimeter Two Refereces []. G.Chrtrd H.Hevi E.B.M. Schultz Sub grph distce grphs defied byedge trsfers Discrete mth []. Byidureg Wu Jixig Meg Hmiltoi Jump grphs Discrete mthemtics []. Girish.A d R.Murli Hmiltoi lcebility i clss of -Regulr Grphs IOSR Jourl of Mthemtics Volume Issue Nov.- Dec. pp 7-. []. G.Mjuth R.Murli d S.K.Rjedr Hmiltoi Lcebility i the Modified Brick Product of Odd Cycles Itertiol Jourl of Grph Theory submitted. []. G.Mjuth d R.Murli Hmiltoi Lcebility i the Brick Product C+r Publish reserch pper i Itertiol Jourl of Grph TheoryGBS Publishers submitted. 6 Pge