K-Even Edge-Graceful Labeling of Some Cycle Related Graphs

Transcription

1 Iteratoal Joural of Egeerg Scece Iveto ISSN (Ole): 9 7, ISSN (Prt): Volume Issue 0ǁ October. 0 ǁ PP.0-7 K-Eve Edge-Graceful Labelg of Some Cycle Related Grahs Dr. B. Gayathr, S. Kousalya Dev, M.Sc., M.Phl., Ph.D., Pgdca., M.C.A., Assocate Professor Ad Research AdvsorPost Graduate & Research Deartmet Of Mathematcs Peryar E.V.R.College (Autoomous) Truchraall-0 0. Tamladu. Ida. ABSTRACT: I 985, Lo[] troduced the oto of edge-graceful grahs. I [], Gayathr et al., troduced the eve edge-graceful grahs. I [8], S-M Lee, Kuo-Jye Che ad Yug-Ch Wag troduced the k-edgegraceful grahs.we troduced k-eve edge-graceful grahs. I ths aer, we vestgate the k-eve edgegracefuless of some cycle related grahs. KEYWORDS: k-eve edge-graceful labelg, k-eve edge-graceful grahs. AMS(MOS) subject classfcato: 05C78. I. INTRODUCTION All grahs ths aer are fte, smle ad udrected. Terms ot defed here are used the sese of Harary[5]. The symbols V(G) ad E(G) wll deote the vertex set ad edge set of a grah G. The cardalty of the vertex set s called the order of G deoted by. The cardalty of the edge set s called the sze of G deoted by q. A grah wth vertces ad q edges s called a (, q) grah. I 985, Lo[] troduced the oto of edge-graceful grahs. I [], Gayathr et al., troduced the eve edge-graceful grahs ad further studed. I [8], S-M Lee, Kuo-Jye Che ad Yug-Ch Wag troduced the k-edge-graceful grahs. We have troduced k-eve edge- graceful grahs. Defto.: k-eve edge-graceful labelg (k-eegl) of a (, q) grah G(V, E) s a jecto f from E to {k, k, k +,..., k + q } such that the duced mag f defed o V by f ( x) = (f(xy)) (mod s) take over all edges xy are dstct ad eve where s = max{, q} ad k s a teger greater tha or equal to. A grah G that admts k-eve edge-graceful labelg s called a k-eve edge-graceful grah (K-EEGG). Remark.: -eve edge-graceful labelg s a eve edge-graceful labelg. The defto of k-edge-graceful ad k-eve edge-graceful are equvalet to oe aother the case of trees. The edge-gracefuless ad eve edge-gracefuless of odd order trees are stll oe. The theory of -eve edge-graceful s comletely dfferet from that of k- eve edge-graceful. For examle, tree of order s -eve edge-graceful but ot -eve edge-graceful. I ths aer we vestgate the k-eve edge-gracefuless of some cycle related grahs. Throughout ths aer, we assume that k s a ostve teger greater tha or equal to.. Pror Results:.Theorem : If a (, q) grah G s k-eve edge-graceful wth all edges labeled wth eve umbers ad q the G s k-edge-graceful..theorem : If a (, q) grah G s k-eve edge-graceful whch all edges are labeled wth eve umbers ad q the q ( q k ) ( ) (m o d ). 0 (m o d ) f s o d d Further q(q + k ) (m o d ) f s e v e Page

2 K-Eve Edge-Graceful Labelg Of.Theorem : If a (, q) grah G s k-eve edge-graceful whch all edges are labeled wth eve umbers ad q the 0, or (mod )..Theorem : If a (, q) grah G s a k-eve edge-graceful tree of odd order the k = (l ) + where l s ay odd ostve teger ad hece k (mod ). 5. Observato : We observe that ay tree of odd order has the sum of the labels cogruet to 0 (mod ).. Theorem : If a (, q) grah G s a k-eve edge-graceful tree of eve order wth 0 (mod ) the k = Further (l ) + where l s ay ostve teger. (m o d ) f l s o d d k (m o d ) f l s e v e. MAIN RESULTS Defto. Let C deote the cycle of legth. The the jo of {e} wth ay oe vertex of C s deoted by C {e}. I ths grah, = q = +. Theorem. k z m o d Proof The grah C {e} of eve order s k-eve-edge-graceful for all, where 0 z, (mod ) ad. Let {v, v,..., v, v} be the vertces of C {e}, the edges e = ad e + = v v (see Fgure )., v, v for ; e = v, v v - v e - e v e e e + v v v Fgure : C {e} wth ordary labelg Frst, we label the edges as follows: For k, ad s odd, f e = k +. Whe s eve, we label the edges as follows: For k z m o d, 0 z, Page

3 K-Eve Edge-Graceful Labelg Of For For For z 7 k fo r f e = z 7 k fo r. k m o d, f e = k + +. k m o d, f e = k k z m o d, z, z 9 k fo r f e = z 9 k fo r. f e = k w h e k 0 m o d k z w h e k z m o d a d z. The the duced vertex labels are as follows: Case : k z m o d Case : Case : k f v k, 0 z z 7 z 5 fo r = z 7 z 5 fo r. + m o d = ; f v + m o d = for. 0 Case : k z m o d, z = for. Page

4 K-Eve Edge-Graceful Labelg Of z 9 z 7 fo r < = z 9 z 7 fo r. For k z m o d, 0 z, f v = 0. Therefore, f V = {0,,,..., s }, where s = max{, q} = +. So, t follows that the vertex labels are all dstct ad eve. Hece, the grah C {e} of eve order s k-eve-edge-graceful for all k z m o d, where 0 z, (mod ) ad. For examle, cosder the grah C {e}. Here = ; s = max {, q} = ; s = 8. The -EEGL of C {e} s gve Fgure Fgure : -EEGL of C {e} 0 For examle, cosder the grah C 7 {e}. Here = 7; s = max {, q} = 8; s =. The 5-EEGL of C 7 {e} s gve Fgure Fgure : 5-EEGL of C 7 {e} 0 Page

5 K-Eve Edge-Graceful Labelg Of Defto. [8] For, a cycle (of order ) wth oe chord s a smle grah obtaed from a -cycle by addg a chord. Let the -cycle be v v... v v. Wthout loss of geeralty, we assume that the chord jos v wth ay oe v, where. Ths grah s deoted by C (). For examle 5 C meas a grah obtaed from a - cycle by addg a chord betwee the vertces v ad v 5. I ths grah, q = +. Theorem. The grah C (), ( > ),, cycle wth oe chord of odd order s k-eve-edge-graceful for all q q k z m o d, where 0 z. Proof Let {v, v, v,..., v, v +,..., v } be the vertces of C (), the edges e = v, v for ; e = v, v ad e + = v v, (see Fgure ). The chord coectg the vertex v wth v, ( ) s show Fgure. For ths grah, = ad q = +., v v e e v e - e v - v e - e v - e + v e e + v + e + e v v + Fgure : C () wth ordary labelg Frst, we label the edges as follows: For k ad, f e = f e = k w h e s o d d k w h e s eve. k w h e k 0 m o d q k z w h e k z m o d q, z q. The the duced vertex labels are as follows: Case I: (mod ) ad q Case : k z m o d, 0 z q + z 7 z 5 fo r = z 7 z 7 fo r. 5 Page

6 K-Eve Edge-Graceful Labelg Of Case : q q k m o d = for. Case : q q 0 q k z m o d, z Case II: (mod ) ad z 9 z 7 fo r = z 9 z 9 fo r. q q Case : k z m o d, 0 z Case : Case : z 7 z 5 fo r = z 7 z 7 fo r. q q k m o d = for. q q 8 q k z m o d, z = z 9 z 7 fo r z 9 z 9 fo r. Therefore, f V are all dstct ad eve. Hece, the grah {0,,,..., s }, where s = max {, q} = +. So, t follows that the vertex labels C (), ( > ),, cycle wth oe chord of odd order s k-eve-edge-graceful for all k z m o d q 0 z. For examle, cosder the grah C 9 5. Here = 9; s = max {, q} = 0; s = 0. q, Page

7 The 5-eve-edge-graceful labelg of C 9 5 s gve Fgure 5. K-Eve Edge-Graceful Labelg Of Fgure 5: 5-EEGL of C 9 (5) For examle, cosder the grah C (). Here = ; s = max {, q} = ; s =. The -EEGL of C () s gve Fgure Theorem.5 The grah 7 5 Fgure : -EEGL of C () C, ( ),, cycle wth oe chord of eve order s k-eve-edge-graceful for all k z m o d q, where 0 z q ad 0 m o d. Proof Let the vertces ad edges be defed as Theorem... Frst, we label the edges as follows: For k, f e = k + ; f e = k f e = k + for. For k ad, k w h e s o d d f e = k w h e s eve. 7 Page

8 K-Eve Edge-Graceful Labelg Of f e = k w h e k 0 m o d q k z w h e k z m o d q, z q. The the duced vertex labels are as follows: Case : k 0 (mod q) f v f v = ; f v =. Case : k z (mod q), z = 0. 8 fo r = fo r. q For k z (mod q), z q, f v For, = z + 8 whe =,. = z + For k q z 8 fo r z z 0 fo r z = z fo r z z 8 fo r z. q m o d, f v = 0 Case : k q f v f v 0 fo r = 8 fo r. q + m o d = ; f v =. =. 8 Page

9 K-Eve Edge-Graceful Labelg Of fo r < = 0 w h e fo r. k z m o d q q +, z q Case : f v = z + 0 whe =,. = z. For k z (mod q), For k q (mod q), q z q, z 0 fo r z z fo r z = z 8 fo r z z 0 fo r z. fo r = 0 fo r. Therefore, f V 0,,,..., s, where s = max{, q} = +. So, t follows that the vertex labels are all dstct ad eve. Hece, the grah C, ( ),, cycle wll oe chord of eve order s k-eve-edge-graceful for all k z (mod q), where 0 z q ad 0 (mod ). For examle, cosder the grah C (5), Here = ; s = max{, q} = 7; s =. The 8-EEGL of C (5) s gve Fgure Fgure 7: 8-EEGL of C (5) 9 Page

10 K-Eve Edge-Graceful Labelg Of Defto. The crow C K s the grah obtaed from the cycle C by attachg edat edge at each vertex of the cycle ad s deoted by Theorem.7 The crow grah C, where 0 z ad 0 (mod ). Proof C. I ths grah, = q =. of eve order s k-eve-edge-graceful for all k z m o d, For ths grah, = q =. Let v, v, v,..., v ad v, v, v,..., v be the vertces ad edat vertces of C resectvely. v e v e e v e v v e v e - e v e v - v e v e - e v e v + Fgure 8: C wth ordary labelg The edges are defed by e = ad v, v for ; e = v, v e = v, v for (see Fgure 8). Frst, we label the edges as follows: For k, f e = k + 5 for f e = k f e = k + ( + ) for. The the duced vertex labels are as follows: Case : k 0 m o d 0 Page

11 K-Eve Edge-Graceful Labelg Of = 0 w h e, 0 fo r. Case : Case : m o d, k z z ad z s odd 5 z z 0 fo r = 5 z z 0 fo r. m o d, k z z ad z s eve z z 0 fo r = z z 0 fo r. The edat vertces wll have the labels (mod ) of the edges cdet o them. Clearly, the vertex labels are all dstct ad eve. Hece, the crow grah C, for all k z m o d, where 0 z ad 0 (mod ). of eve order s k-eve-edge-graceful For examle, cosder the grah C. Here = q = ; s = max{, q} = ; s =. The -eve-edge- graceful labelg of C s gve Fgure Fgure 9: -EEGL of C + Page

12 Theorem.8 K-Eve Edge-Graceful Labelg Of The crow grah C, of eve order s k-eve-edge-graceful for all k z m o d, where 0 z, (mod ) ad. Proof Let the vertces ad edges be defed as Theorem... The edge labels are also same as Theorem... The the duced vertex labels are as follows: Case : k 0 (mod ) 0 w h e, = 0 fo r. Whe z s odd, the duced vertex labels are gve below: Case : k z (mod ), z Case : k z (mod ), + 5 z z 0 fo r = 5 z z 0 fo r. + z + 5 z z 0 fo r = 5 z z 8 0 fo r. Case : k z (mod ), + 5 z 5 z z 8 0 fo r + = 5 z z 0 fo r. Whe z s eve, the duced vertex labels are gve below: Case 5: k z m o d, z Case : k z + z z 0 fo r = z z 0 fo r. + + m o d, z Page

13 K-Eve Edge-Graceful Labelg Of Case 7: z z 0 fo r = z z 8 0 fo r. k z m o d + 5, z z z 8 0 fo r = z z 0 fo r. The edat vertces wll have the labels (mod ) of the edges cdet o them. Clearly, the vertex labels are all dstct ad eve. Hece, the crow grah C, for all k z m o d, where 0 z, (mod ) ad. of eve order s k-eve-edge-graceful For examle, cosder the grah C. Here = q = ; s = max {, q} = ; s = 8. The 5-eve-edge- 7 graceful labelg of C s gve Fgure Theorem Fgure 0: 5-EEGL of C The crow grah C, ( 5) of eve order s k-eve-edge-graceful for all k z (mod ), where 0 z, (mod ) ad. Proof Let the vertces ad edges be defed as Theorem.7. The edge labels are also same as Theorem.7. The the duced vertex labels are as follows: Case : k 0 (mod ) Page

14 K-Eve Edge-Graceful Labelg Of 0 w h e, f v = 0 fo r. Whe z s odd, the duced vertex labels are gve below: Case : k z (mod ), z Case : k z (mod ), + 5 z z 0 fo r = 5 z z 0 fo r z + 5 z z 0 fo r = 5 z z 8 0 fo r. Case : k z (mod ), + z 5 z z 8 0 fo r = 5 z z 0 fo r. Whe z s eve, the duced vertex labels are gve below: Case 5: k z (mod ), z Case : k z (mod ), + z z 0 fo r = z z 0 fo r. +5 z + z z 0 fo r = z z 8 0 fo r. Case 7: k z (mod ), + z z z 8 0 fo r = z z 0 fo r. Page

15 K-Eve Edge-Graceful Labelg Of The edat vertces wll have the labels (mod ) of the edges cdet o them. Clearly, the vertex labels are all dstct ad eve. Hece, the crow grah C, ( 5) of eve order s k-eve-edge-graceful for all k z (mod ), where 0 z, (mod ) ad. For examle, cosder the grah C. Here = q = 0; s = max {, q} = 0; s = 0. 5 The 0-eve-edge-graceful labelg of C s gve Fgure Fgure : 0-EEGL of C + 5 Defto.0 A grah H m G s obtaed from a grah G by relacg each edge wth m arallel edges. Theorem. The grah H (P ), ( ) of eve order s k-eve-edge-graceful for all k z (mod ), where 0 z ad s eve. Proof Let {v, v,..., v } be the vertces of H (P ). Let the edges e j of e j = (v, v + ) for, j (see Fgure ). H P be defed by v v v v - v e e e - e e e - e e e - e e e - Fgure : H (P ) wth ordary labelg 5 Page

16 For ths grah, = ; q = ( ). Frst, we label the edges as follows: For k, K-Eve Edge-Graceful Labelg Of k j 5 fo r, s o d d a d j f e j = k j ( ) fo r, s eve a d j. The the duced vertex labels are as follows: Case : k 0 (mod ) f v f v = ( ). 9 fo r = 7 fo r. = ( ). Case : k z (mod ), z f v = (z ). Subcase (): k (mod ) f ( v ) f ( v ) Subcase (): f ( v ) = ( ) for. = [ + (z )]. k z (mod ), z z fo r z = z 5 fo r z. = [ + (z )]. Subcase (): k z (mod ), f ( v ) + z z 5 fo r z = z fo r z. = [(z ) ]. Therefore, f V {0,,,..., s }, where s = max {, q} = ( ). So, t follows that the vertex labels are all dstct ad eve. Hece, the grah H (P ), ( ) of eve order s k-eve-edge-graceful for all k z (mod ), where 0 z ad s eve. For examle, cosder the grah H 8 (P 8 ). Here = 8; q = 5; s = max {, q} = 5; s =. The -EEGL of H 8 (P 8 ) s gve Fgure. Page

17 K-Eve Edge-Graceful Labelg Of Fgure : -EEGL of H 8 (P 8 ) REFERENCES []. G.S. Bloom ad S.W. Golomb, Alcatos of umbered udrected grahs, Proc. IEEE, 5(977) []. G.S. Bloom ad S.W. Golomb, Numbered comlete grahs, uusual rulers, ad assorted alcatos, Theory ad Alcatos of Grahs, Lecture Notes Math.,, Srger-Verlag, New York (978) 5 5. []. J.A. Galla, A dyamc survey of grah labelg, The electroc joural of Combatorcs (009), #DS. []. B. Gayathr, M. Durasamy, ad M. Tamlselv, Eve edge graceful labelg of some cycle related grahs, It. J. Math. Comut. Sc., (007) [5]. F. Harary, Grah Theory Addso Wesley, Readg Mass (97). []. S. Lo, O edge-graceful labelgs of grahs, Cogr. Numer., 50 (985). [7]. W.C. Shu, M.H. Lg, ad R.M. Low, The edge-graceful sectra of coected bcyclc grahs wthout edat, JCMCC, (008) [8]. S-M Lee, Kuo-Jye Che ad Yug-Ch Wag, O the edge-graceful sectra of cycles wth oe chord ad dumbbell grahs, Cogressus Numeratum, 70 (00) Page