Some Results on E - Cordial Graphs

Transcription

1 Intrnational Journal of Mathmatics Trnds and Tchnology Volum 7 Numbr 2 March 24 Som Rsults on E - Cordial Graphs S.Vnkatsh, Jamal Salah 2, G.Sthuraman 3 Corrsponding author, Dpartmnt of Basic Scincs, Collg of Applid Scincs, A Sharqiyah Unirsity, P.O. Box.42, P.C.4, Sultanat of Oman. 2 Dpartmnt of Basic Scincs, Collg of Applid Scincs, A Sharqiyah Unirsity, P.O. Box.42, P.C.4, Sultanat of Oman. 3 Dpartmnt of Mathmatics, Anna Unirsity, Guindy campus, Chnnai 25, India. Abstract Lt G(V, E) b a finit simpl graph. Lt φ b a function from th dg st E to {, }. For ach rtx V, dfin φ() = {φ(u): u E}(mod 2). Th function φ is calld an E cordial labling of G, if th numbr of dgs labld and th numbr of dgs labld diffrs by at most on and th numbr of rtics labld and th numbr of rtics labld diffrs by at most on. A graph that admits E cordial labling is said to E - Cordial. In this papr, w pro that Odd Snaks and C m (t), that is on rtx union of t copis of C m, for t n and m odd ar E - Cordial. Kywords Snaks, On rtx union of cycls, E - Cordial labling, E - Cordial graphs.. INTRODUCTION Lt G(V, E) b any finit simpl graph. For any graph-thortic notation w follow Wst [].A rtx labling or aluation of a graph G is an assignmnt f of labls to th rtics of G, that inducs for ach dg = xy, a labl dpnding on th rtx labls f(x) and f(y). Yilmaz and Cahit [4] introducd dg-cordial labling as a wakr rsion of dg gracful labling. Lt φ b a function from th dg st Eto {,}. For ach rtx V, dfin φ() = {φ(u) u E}(mod 2). Th function φ is calld an E cordial labling of G, if th numbr of dgs labld and numbr of dgs labld diffr by at most on and th numbr of rtics labld and th numbr of rtics labld diffr by at most on. A graph that admits an E cordial labling is said to b E Cordial graph. It is obsrd that graphs with n 2 (mod 4) rtics cannot b E - Cordial. For mor rlatd rsults on E - Cordial graphs on can rfr Gallian [2]. 2. Odd Snaks ar E - Cordial. Rcall that snaks ar th graphs whos block cut point graph is a path and furthr odd snak is a snak whos blocks ar isomorphic to a cycl C of lngth m odd(= 2n + ). Rfr Figur.. In this sction w pro that odd snak is E - Cordial. 2.. Thorm. Odd Snak is dg Cordial. Proof: LtG b an odd snakas gin in figur. Thn by dfinition, G contains r blocks B, i r and ach block B of G is isomorphic to a cycl of lngth m(= 2n + ), that is, C,,,,,,,. For conninc of th labling, for i r, lt, =,.Thus from th abo, w obsr that th blocks B and B ha a common cut rtx, for 2 i r. Arrang th dgs of all th blocks B of G in th following ordr {,,,,,,, } as shown in figur.. Lt E and E dnot th st of all dgs assignd th labl and rspctily. In th sam way, lt V and V dnot th st of all rtics assignd th labl and rspctily. ISSN: Pag 2

2 Intrnational Journal of Mathmatics Trnds and Tchnology Volum 7 Numbr 2 March 24 Dfin th labling φ E(G) {,}corrsponding to th dgs of G in following cass,, if r is odd Cas :. If m = 3, thn for i r and j 3, dfin φ, =, if r is n Th inducd rtx labling is as follows,, =,, =,, =. For 2 i r, i n,, =,, =. For 3 i r, i odd,, =,, =. Hnc, if m = 3, it is obsrd that, if th numbr of blocks r is odd, thn E = E + and V = V + and if r is n, thn E = E and V = V + Cas : 2. If m = 4k +, thn for i r and j 2n +, Dfin φ, = (), if r is odd (), if r is n Th inducd rtx labling is as follows,, =., if j 2n, j odd For i r,, =, if 2 j 2n,j n. Hnc if m = 4k +, it is obsrd that, if th numbr of blocks r is odd thn, E = E + and V = V + and if r is n, thn E = E and V = V +. Cas : 3. If m = 4k + 3, thn for i r and j 2n +, Dfin φ, = (), if r is odd (), if r is n Th inducd rtx labling is as follows,, =,, =, if 3 i r,i odd and, =, if 2 i r, i n, if 2 j 2n, j n For i r, and i is odd,, =, if 3 j 2n,j odd, if 2 j 2n,j n For 2 i r, and i is n,, =, if 3 j 2n, j odd If m = 4k + 3, it is obsrd that, if th numbr of blocks r is odd, thn E = E + and V = V + and if r is n, thn E = E and V = V +. Hnc it is clar that G is E - Cordial.,n+,n,n,n+,n+2,n+2 2,n+ 2,n+ 2,n+2 2,n 2,n+2 r, n+ r,n r, n+ r,n+2 r,n+2,2,2,,,2n 2,2n r,2 2,2,2n+ 2,2 2,2n+ r,2 =,2n+ 2, 2, 2,2n+ r, r, r,2n r,2n+ r,2n+ Figur.. Odd Snak ISSN: Pag 22

3 Intrnational Journal of Mathmatics Trnds and Tchnology Volum 7 Numbr 2 March Illustrations to Thorm. 2.. Fig.2. E Cordial labling of C 3 snak Fig.3. E Cordial labling of C 5 snak 3. C ( ), ONE VERTEX UNION OF EVEN NUMBER OF ODD CYCLES ARE E - CORDIAL. Rcall that a graph G is said to b on rtx union of graphs G, G, G,, G if G is obtaind by adjoining G, G, G,, G at a common rtx, that is, G = G, for i =,2, 3,, t. In this sction it is prod that C ( ), th on rtx union of tisomorphiccopis of cycls C, for t n and m odd ar E Cordial. 3.. Thorm.C ( ), on rtx union of t isomorphic copis of cycls C, for t n and m odd is E - Cordial. Proof: Lt G b th on rtx union of G, i t, t n, whrg C ( ).For i t,ordr th copis G as G, G,, G. For i t, nam th rtics of G as follows, {,,,,,,,, (= )}, whr is a common rtx. For i t, and nam th dgs of G as follows,, =, ;, =, and, =,,, for 2 j 2n From th abo it is obsrd that for i t, th common rtx is incidnt with th dgs,,, and,,,,,,,,,, inducs th cycl C 2n+ i, i,2 i,2 i,3 i,3 i,2n i,2n+. Rfr Figur. 4. For th conninc of th labling arrang th dgs,, for i t, j 2n + of G as a squnc in th form,,,,,,,,. ISSN: Pag 23

4 Intrnational Journal of Mathmatics Trnds and Tchnology Volum 7 Numbr 2 March 24 t,n+ 2,n+ t,n+2 t,n t,n+2 t,n t,n+4 t,n+3 t,n- t,n+4 t,n- t,2n t,2 t,2n+,n-2,,,2,2 2, 2,2n+ 2,2n,2n+,2n,2n,2n- 2,2,n+4 2,n- 2,n+4 2,n- 2,n+3 2,n+4 2,n 2,n+2 2,n 2,n+2 2,n+ 2,n+,n-2,n+3,n,n+3,n,n+2,n+,n+,n+2 Fig.4. G = G, for i =,2, 3,, t Lt E and E dnot th st of all dgs assignd th labl and rspctily. In th sam way, lt V and V dnot th st of all rtics assignd th labl and rspctily. Dfin th labling φ E(G) {,} corrsponding to th dgs of G in following cass,, if t is odd Cas :. If m = 3, thn for i t and j 3, dfin φ, =, if t is n Th inducd rtx labling is as follows, For i t, i odd,, =,, =. For 3 i t, i n,, =,, =. Hnc, if m = 3, it is obsrd that, E = E and V = V +. Cas : 2. If m = 4k +, thn for i t and j 2n +, Dfin φ, = (), if t is odd (), if t is n Th inducd rtx labling is as follows,, if j 2n,j odd For i t,, =, if 2 j 2n, j n. Hnc if m = 4k +, it is obsrd that E = E and V = V +. Cas : 3. If m = 4k + 3, thn for i t and j 2n +, Dfin φ, = (), if t is odd (), if t is n Th inducd rtx labling is as follows, ISSN: Pag 24

5 Intrnational Journal of Mathmatics Trnds and Tchnology Volum 7 Numbr 2 March 24, if j n For i t, and i is odd,, =, if j odd, if j n For 2 i t, and i is n,, =, if j odd If m = 4k + 3, it is obsrd that, E = E and V = V + and hnc G is E - Cordial. 4. CONCLUSIONS Yilmaz and Cahit ha discussd E Cordial labling of C ( ) for all alus n, whil in this papr E Cordial labling of C ( ), for t n and m odd ha bn instigatd. REFERENCES [] D.B. Wst. Introduction to Graph Thory, Prntic Hall of India, 2. [2] J.A. Gallian, A dynamic sury of graph labling, Th Elctronic Journal of Combinatorics,# DS6, 23. [3] I. Cahit, Cordial Graphs : A wakr rsion of gracful and harmonious graphs, Ars Combinatoria, 23, pp. 2 27, 987. [4] R. Yilmaz and I. Cahit E-Cordial graphs, Ars Combinatoria, 46, pp , 997. ISSN: Pag 25