Edge Product Cordial Labeling of Some Cycle Related Graphs

Op Joua o Dsct Mathmatcs, 6, 6, 68-78 http://.scp.o/joua/ojdm ISSN O: 6-7643 ISSN Pt: 6-7635 Ed Poduct Coda Lab o Som Cyc Ratd Gaphs Udaya M. Pajapat, Ntta B. Pat St. Xav s Co, Ahmdabad, Ida Shaksh Vaha Bapu Isttut o Tchooy, Gadhaa, Ida Ho to ct ths pap: Pajapat, U.M. ad Pat, N.B. (6) Ed Poduct Coda Lab o Som Cyc Ratd Gaphs. Op Joua o Dsct Mathmatcs, 6, 68-78. http://dx.do.o/.436/ojdm.6.643 Rcvd: Ju 8, 6 Accptd: Sptmb 6, 6 Pubshd: Sptmb 9, 6 Copyht 6 by authos ad Sctc Rsach Pubsh Ic. Ths ok s csd ud th Catv Commos Attbuto Itatoa Lcs (CC BY 4.). http://catvcommos.o/css/by/4./ Op Accss Abstact Fo a aph G = ( V ( G), E( G) ) hav o soatd vtx, a ucto : E( G) {,} s cad a d poduct coda ab o aph G, th ducd vtx ab ucto dd by th poduct o abs o cdt ds to ach vtx s such that th umb o ds th ab ad th umb o ds th ab d by at most ad th umb o vtcs th ab ad th umb o vtcs th ab aso d by at most. I ths pap, dscuss d poduct coda ab o som cyc atd aphs. Kyods Gaph Lab, Ed Poduct Coda Lab. Itoducto W b th smp, t, udctd aph G ( V ( G), E( G) ) vtx h V ( G ) ad ( ) v( G ) ad ( ) = hav o soatd E G dot th vtx st ad th d st spctvy, E G dot th umb o vtcs ad ds spctvy. Fo a oth tmooy, oo Goss []. I th pst vstatos, C dots cyc aph th vtcs. W v b summay o dtos hch a usu o th pst ok. Dto. A aph ab s a assmt o ts to th vtcs o ds o both subjct to th cta codtos. I th doma o th mapp s th st o vtcs (o ds) th th ab s cad a vtx (o a d) ab. Fo a xtsv suvy o aph ab ad bboaphy cs, to Gaa []. DOI:.436/ojdm.6.643 Sptmb 9, 6

U. M. Pajapat, N. B. Pat Dto. Fo a aph G, th d ab ucto s dd as : E( G) {,} ad ducd vtx ab ucto * : V ( G) {,} s v as,,, * a th ds cdt to th vtx v th ( v) = ( ) ( ) ( ). * Lt v ( ) b th umb o vtcs o G hav ab ud ad ( ) a b th umb o ds o G hav ab ud o =,. s cad a d poduct coda ab o aph G v ( ) v ad ( ). A aph G s cad d poduct coda t admts a d poduct coda ab. Vadya ad Baasaa [3] toducd th cocpt o th d poduct coda ab as a d aaou o th poduct coda ab. Dto 3. Th h W ( > 4) s th aph obtad by add a vtx jo to ach o th vtcs o C. Th vtx s cad th apx vtx ad th vtcs cospod to C a cad m vtcs o W. Th ds jo m vtcs a cad m ds. Dto 4. Th hm H s th aph obtad om a h W by attach a pdat d to ach o th m vtcs. Dto 5. Th cosd hm CH s th aph obtad om a hm H by jo ach pdat vtx to om a cyc. Th cyc obtad ths ma s cad a out cyc. Dto 6. Th b Wb s th aph obtad by jo th pdat vtcs o a hm H to om a cyc ad th add a pdat d to ach o th vtcs o th out cyc. Dto 7. Th Cosd Wb aph CWb s th aph obtad om a b aph Wb by jo ach o th out pdt vtcs coscutvy to om a cyc. Dto 8. [4] Th Suo aph SF s th aph obtad by tak a h th th apx vtx v ad th coscutv m vtcs v, v,, v ad addtoa vtcs,,, h s jod by ds ad v + ). Dto 9. [4] Th Lotus sd a cc LC s a aph obtad om a cyc C : uu uu ad a sta aph K, th ct vtx v ad d vtcs v, v,, v by jo ach v to u ad u+ ). Dto. [5] Dupcato o a vtx v o a aph G poducs a aph G by add a vtx v such that N( v ) = N( v). I oth ods, v s sad to b a dupcato o v a th vtcs hch a adjact G a aso adjact G. Dto. [5] Dupcato o a vtx v k by a d = vv k k a aph G poducs a aph G such that N( v k) = { vk, v k} ad N( v k) = { vk, v k}. Dto. Th Fo aph F s th aph obtad om a hm H by jo ach pdat vtx to th apx o th hm H.. Ma Rsut Thom. Cosd b aph CWb s ot a d poduct coda aph. Poo. Lt v b th apx vtx ad v, v, v3,, v b th coscutv m vtcs 69

U. M. Pajapat, N. B. Pat o W. Lt vv + a a a vv,,, =, o ach =,,, (h subscpts a moduo ). Lt =. Lt v v v b th vtcs cospod, v, v 3,, v a a spctvy. Fom th d by jo v ad v. Fom th cyc by cat a a a a a a a a a = vv, = vv,, = vv. Th sut aph s cad a hm ds ( ) ( 3 ) ( ) b b b a a a aph. Lt v, v,, v b th vtcs cospod, v,, v spcc a b tvy. Fom th d by jo v ad v. Fom th cyc by cat th ds v b b b b b b b b = vv, = vv,, = vv. Th sut aph s a cosd b aph ( ) ( ) ( ) 3 CWb. Thus E ( CWb ) = 6 ad V ( CWb ) 3 mapp : ( ) {,} E CWb cosd oo to cass: = +. W a ty to d th Cas : I s odd th od to satsy th d codto o d poduct coda aph t s ssta to ass ab to 3 ds out o 6 ds. So ths cotxt, th ds th ab v s at ast 3 + vtcs th ab ad at most 3 vtcs th ab out o 3 + vtcs. Tho v ( ) ( ) v. So CWb s ot a d poduct coda aph o odd. Cas : I s v th od to satsy th d codto o d poduct coda aph t s ssta to ass ab to 3 ds out o 6 ds. So ths cotxt, th ds th ab v s at ast 3 + vtcs th ab ad at most 3 vtcs th ab out o 3 + vtcs. Fom both th cass v ( ) v, so CWb s ot a d poduct coda aph. Thom. Lotus sd cc LC s ot a d poduct coda aph. Poo. Lt v b th apx vtx ad v, v, v3,, v b th coscutv vtcs o a a a sta aph K,. Lt = vv o ach =,,,. Lt v, v,, v b th vtcs cospod, v, v 3,, v spctvy. Fom th cyc by cat th a a a a a a a a a ds ( = vv ), ( = vv 3 ),, ( = vv ). Lt a a = vv ad = vv +. Th sut aph s Lotus sd cc Lc. Thus E ( LC ) = 4 ad V ( LC ) = +. W a ty to d th mapp : ( ) {,} E LC. I od to satsy a d codto o d poduct coda aph t s ssta to ass ab to ds out o 4 ds. So ths cotxt, th ds th ab v s to at ast + vtcs th ab ad at most vtcs th ab out o + vtcs. Tho v ( ) v ( ) So, 3. LC s ot a d poduct coda aph. Thom 3. Suo aph SF s a d poduct coda aph o 3. Poo. Lt SF b th suo aph, h v s th apx vtx, v, v,, v b th coscutv m vtcs o W ad,,, b th addtoa vtcs h s jod ad v + ). Lt,, 3,, b th coscutv m ds o th W.,,, a th cospod ds jo apx vtx v to th vtcs v, v,, v o th cyc. Lt o ach, b th ds jo to v ad b th ds jo + ). Thus E ( SF ) 4 = ad 7

U. M. Pajapat, N. B. Pat ( ) = +. D th mapp : ( ) {,} { } = ( ) V SF E SF as oos:,,,,3,, ; = {, }, =,,3,,. I v o th abov dd ab patt hav, v ( ) { v, v, v,, v} ( ) = {,,, }. So, v ( ) = v + = + ad ( ) ( ) v ad ( ) = ad v = =. Thus v. Thus admts a d poduct coda o SF. Hc, SF s a d poduct coda aph. Iustato. Gaph SF 6 ad ts d poduct coda ab s sho Fu. Thom 4. Th aph obtad om dupcato o ach o th vtcs o =,,, by a vtx th suo aph SF s a d poduct coda aph ad oy s v. Poo. Lt SF b suo aph, h v s th apx vtx, v, v,, v b th coscutv m vtcs o W ad,,, b th addtoa vtcs h s jod ad v + ). Lt,, 3,, b th coscutv m ds o th W.,,, a th cospod ds jo th apx vtx v to th vtcs v, v,, v o th cyc. Lt o ach, b th ds jo ad b th ds jo + ( mod ). Lt G b th aph obtad om SF by dupcato o th vtcs,,, by th vtcs u, u,, u spctvy. Lt o ach, b th ds jo u ad b th ds jo u + ). Thus E( G) = 6 ad V ( G) = 3+. W cosd th oo to cass: Cas : I s odd, d th mapp : E( G) {,} od to satsy d codto o d poduct coda aph t s ssta to ass ab to 3 ds out 3 o 6 ds. So ths cotxt, th ds th ab v s at ast + Fu. SF6 ad ts d poduct coda ab 7

U. M. Pajapat, N. B. Pat vtcs th ab ad at most 3 Tho v ( ) v ( ) vtcs th ab out o 3 + vtcs.. So th dupcato o vtx by vtx u suo aph s ot d poduct coda o odd. Cas : I s v, d th mapp : E( G) {,} as oos: {, }, =,,3,, ; {, }, =,,3,, ; ( ) = {, }, = +, +,, ; {, }, =,,3,,. I v o th abov dd ab patt hav, v ( ) = v, v, v,, v,,,, ad v ( ) = u, u,, u,,,,. + + 3 So v ( ) = v + = + ad ( ) = = 3. Thus v ( ) v ad. Thus admts a d poduct coda ab o G. So, G s a ( ) ( ) d poduct coda o v. Iustato. Gaph G obtad om SF 6 by dupcato o ach o th vtcs,, 3, 4, 5, 6 by vtcs u, u, u 3, u 4, u 5, u 6 ad ts d poduct coda ab s sho Fu. Thom 5. Th aph obtad om dupcato o ach o th vtcs o =,,, by a ds th suo aph SF s a d poduct coda aph. Fu. G ad ts d poduct coda ab. 7

U. M. Pajapat, N. B. Pat Poo. Lt SF b suo aph, h v s th apx vtx, v, v,, v b th coscutv m vtcs o W ad,,, b th addtoa vtcs h s jod ad v + ). Lt,, 3,, b th coscutv m ds o W.,,, a th cospod ds jo apx vtx v to th vtcs v, v,, v o th cyc. Lt o ach, b th ds jo ad b th ds jo + ( mod ). Lt G b th aph obtad om SF by dupcato o th vtcs,,, by cospod ds,,, th vtcs u ad u such that u ad u jo to th vtx. Lt o ach, b th ds jo th vtcs u ad u ad o ach, b th ds jo u to ad b th ds jo u to. Thus E( G) = 7 ad V ( G) = 4+. W cosd th oo to cass: Cas : I s odd, d th mapp : E( G) {,} as oos: =, =,,3,, ; =, =,3,, ; + ( ) {,,, u, u }, =,,3,, ; = + + 3 + 5 {, }, =,,,, ; = ; + 3 + 5 + 7 {,,, u, u }, =,,,,. I v o th abov dd ab patt hav, ( ),,,,,,,,,,,,, v = v v v v v u+ u+ u u+ u+ u, +, +,, + + + + + + + + ad v ( ) = v, v3,, v, u, u,, u+, u, u,, u+,,,, +. So, 7 v ( ) = v + = + ad =, ( ) 7 =. Cas : I s v, d : E( G) {,} as oos: =, =,,3,, ; =, =,3,4,, ; {, }, =,,3,, ; + ( ) {, u, u }, =,,3,, ; = + + 4 {, }, =,,,, ; = ; + + 4 + 6 {, }, =,,,, ; + 4 + 6 + 8 {, u, u }, =,,,,. 73

U. M. Pajapat, N. B. Pat I v o th abov dd ab patt hav, v ( ) = v, v, v, v,, v, u+, u+,, u, u+, u+,, u,,,, + + + + + + + + ad v ( ) = v, v3,, v, u, u,, u+, u, u,, u+,,,,. So, 7 v ( ) = v + = + ad ( ) = ( ) =. v v. Thus, admts a Fom both th cass ( ) ( ) ad ( ) ( ) d poduct coda ab o G. So th aph G s a d poduct coda aph. Iustato 3. Gaph G obtad om SF 5 by dupcato o ach o th vtcs,, 3, 4, 5 by cospod ds,, 3, 4, 5 ad ts d poduct coda ab s sho Fu 3. Thom 6. Th aph obtad by dupcato o ach o th vtcs th suo aph SF s ot a d poduct coda aph. Poo. Lt SF b th suo aph, h v s th apx vtx, v, v,, v b th coscutv vtcs o th cyc C ad,,, b th addtoa vtcs h s jod ad v + ). Lt,, 3,, b th coscutv ds o th cyc C.,,, a th cospod ds jo th apx vtx v to th vtcs v, v,, v o th cyc. Lt o ach, b th d jo ad b th d jo + ). Lt G b th aph obtad om SF by dupcato o ach o th vtcs v, v, v,, v,,,, by th vtcs v, v, v,, v,,,, spctvy. Fu 3. G ad ts d poduct coda ab. 74

U. M. Pajapat, N. B. Pat a a a Lt,,, b b b =,,,.,,, 5 o b th ds jo th vtx to th adjact vtx o o a th ds jo th vtx v to th adjact vtx c c c v o =,,, ad,,, a th ds jo th vtx v to th E G = ad V ( G) = 4+. adjact vtx o c th suo aph. Thus ( ) W a ty to d : E( G) {,}. I od to satsy th d codto o d poduct coda aph, t s ssta to ass ab to 6 ds out o ds. So ths cotxt, th d th ab 4 + 4 + v s at ast + vtcs th ab ad at most vtcs th ab out o 4 + vtcs. Tho, v ( ) v. So th aph G s ot a d poduct coda aph. Thom 7. Th aph obtad by subdvd th ds v ad v + (subscpts a mod ) o a =,,..., by a vtx th suo aph SF s a d poduct coda. Poo. Lt SF b th suo aph, h v s th apx vtx, v, v,, v b th coscutv m vtcs o W ad,,, b th addtoa vtcs h s jod ad v + ). Lt,, 3,, b th coscutv m ds o.,,, a th cospod ds jo apx vtx v to th vtcs v, v,, v o th cyc. Lt G b th aph obtad om SF by subdvd th ds v ad v + ) o a =,,, by a vtx u ad u spctvy. Lt o ach, b th ds jo u ad b th ds jo u to ( mod v+ ). Lt o ach, b th ds jo to u ad b th ds jo to u. Thus, E( G) = 6 ad V ( G) = 4+. W cosd th oo to cass: : E G, as oos: Cas : I s odd, d ( ) { } { },, =,,3,, ; + 3 + 5 + 7 =, =,,,, ; + + 3 + 5 =, =,,,, ; ( ) = + =, =,,3,, ; =, =,,3,, ; {, }, =,,3,,. I v o th abov dd ab patt hav, v ( ) = v, v, v,, v, u+, u+,, u, u, u,, u ad + + + + ( ),,,,,,, v = u u u+ u u u,,,. So v ( ) = v + = + ad = = 3. ( ) ( ) 75

U. M. Pajapat, N. B. Pat Cas : I s v, d : E( G) {,} as oos: {, }, =,,3,, ; + + 4 + 6 {, }, =,,,, ; ( ) = {, }, =,,3,, ; {, }, =,,3,,. I v o th abov dd ab patt hav, v ( ) = v, v, v,, v, u, u,, u, u, u,, u ad + + + + ( ),,,,,,, v = u u u u u u,,,. So v ( ) = v + = + ad ( ) = = 3. Fom both th cass v ( ) v ad ( ) ( ). Thus admts a d poduct coda ab o G. So th aph G s a d poduct coda aph. Iustato 4. Gaph G obtad om SF 8 by subdvd th ds v ad v + ) o a =,,, 8 by a vtx u ad u spctvy o =,,, 8 ad ts d poduct coda ab s sho Fu 4. Thom 8. Th aph obtad by o aph F by add pdat vtcs to th apx vtx v s a d poduct coda aph. Poo. Th Fo aph F s th aph obtad om a hm H by jo Fu 4. G ad ts d poduct coda ab. 76

U. M. Pajapat, N. B. Pat ach pdat vtx to th apx vtx o th hm H. Lt G b th aph obtad om th o aph F by add pdat vtcs u, u,, u to th apx vtx v. Lt v, v,, v b th m vtcs ad,,, b th pdat vtcs, v,, v spctvy. Lt,, 3,, b th coscutv m ds o W.,,, a th cospod ds jo th apx vtx v to th vtcs v, v,, v o th cyc. Lt a b c = v o ach =,,,. Lt = v o ach =,,,. Lt = uv o ach =,,,. Thus E( G) = 5 ad V ( G) = 3+. W cosd to cass: Cas : I s v, d th mapp : E( G) {,} as oos: {, }, =,,3,, ; a b {, }, =,,3,, ; ( ) = c =, =,,3,, ; c + + 4 + 6 =, =,,,,. I v o th abov dd ab patt hav, v ( ) = v, v, v,, v, u +, u +,, u ad 3 v ( ) =,,,, u, u,, u. So v ( ) = v + = + ad 5 ( ) = ( ) =. Cas : I s odd, d th mapp : E( G) {,} as oos: {, }, =,,3,, ; a b {, }, =,,3,, ; ( ) = c + =, =,,3,, ; c + 3 + 5 + 7 =, =,,,,. I v o th abov dd ab patt hav, v ( ) = v, v, v,, v, u+, u+,, u ad + + 3 + 5 v ( ) =,,,, u, u,, u +. So v ( ) = v = ad =, 5 ( ) =. Fom both th cass v ( ) v ad ( ). Thus, admts a d poduct coda ab o G. So G s a d poduct coda aph. Iustato 5. Gaph G obtad om F 5 by add 5 pdt vtcs u, u, u3, u4, u 5 to th apx vtx v ad ts d poduct coda ab s sho Fu 5. 77

U. M. Pajapat, N. B. Pat Fu 5. G ad ts d poduct coda ab. 3. Cocud Rmaks W vstatd ht suts o th d poduct coda ab o vaous aph atd by a cyc. Sma pobm ca b dscussd o oth aph ams. Ackodmts Th authos a hhy thaku to th aoymous o vauab commts ad costuctv sustos. Th st autho s thaku to th Uvsty Gat Commsso, Ida o suppot hm th Mo Rsach Pojct ud No. F. 47-93/4 (WRO) datd th Mach, 5. Rcs [] Goss, J.L. ad Y, J. (Eds.) (4) Hadbook o Gaph Thoy. CRC Pss, Boca Rato. [] Gaa, J.A. (4) A Dyamc Suvy o Gaph Lab. Th Ectoc Joua o Combatocs, 7, #DS6. http://.combatocs.o [3] Vadya, S.K. ad Baasaa, C.M. () Ed Poduct Coda Lab o Gaphs. Joua o Mathmatca ad computatoa Scc,, 436-45. [4] Poaj, R., Sathsh Naayaa, S. ad Kaa, R. (5) A Not o Dc Coda Gaphs. Past Joua o Mathmatcs, 4, 89-97. [5] Vadya, S.K. ad Pajapat, U.M. (3) Pm Lab th Cotxt o Dupcato o Gaph Emts. Itatoa Joua o Mathmatcs ad Sot Comput, 3, 3-. 78

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