Solution to Some Open Problems on E-super Vertex Magic Total Labeling of Graphs
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1 Aalable a hp://paed/aa Appl Appl Mah ISS: Vol 0 Isse (Deceber 0) pp 04- Applcaos ad Appled Maheacs: A Ieraoal Joral (AAM) Solo o Soe Ope Probles o E-sper Verex Magc Toal Labelg o Graphs G Marh MS Raa Drga G Drga De Depare o Maheacs The Madra College Madra- 60 Talad Ida yellowh@yahooco; sraadrga7@galco; phadrga99@galco Receed: Jaary 0; Acceped: Deceber 7 0 Absrac Le G be a e graph wh p erces ad q edges A erex agc oal labelg s a beco ro VG EG o he cosece egers p+q wh he propery ha or eery V G or soe cosa Sch a labelg s E-sper q : E G A graph G s called E-sper erex agc ads a E-sper erex agc labelg I hs paper we sole wo ope probles ge by Marh Sgaya Kalaa ad Balarsha (Marh e al 0) Keywords: sper erex agc labelg; E-sper erex agc labelg; E-sper erex agc graph 00 AMS Sbec classcao: 0C78 Irodco I hs paper we cosder oly e sple dreced graphs wh order p ad sze q For graph heorec oaos we ollow (Harary 969; Marr ad Walls 0) A labelg o a graph G s a appg ha carres a se o graph elees sally he erces ad edges o a se o bers sally egers May ds o labelgs hae bee sded ad a excelle srey o graph labelg ca be od (Galla 04) Sedláče (Sedláče96) rodced he cocep o agc labelg Sedláče deed a graph o be agc had a edge-labelg wh rage he real bers sch ha he s o he labels arod ay erex eqalled cosa depede o he choce o erex MacDogall e al (MacDogall e al 00) rodced he oo o erex agc oal labelg I G s a 04
2 AAM: Ier J Vol 0 Isse (Deceber 0) 0 e sple dreced graph wh p erces ad q edges he a erex agc oal labelg o G s a beco ro VG EG o he egers p+q wh he propery ha or eery V(G) or soe cosa where V G EG : They sded he basc properes o erex agc oal graphs ad showed soe ales o graphs hag erex agc oal labelg MacDogall e al (MacDogall e al 004) rher rodced he sper erex agc oal labelg They called a erex agc oal labelg s sper : VG p Swaaha ad Jeyah (Swaaha ad Jeyah 00) rodced a cocep wh he ae sper erex agc oal labelg b wh dere oo They called a erex agc oal labellg o be sper : EG q To aod coso Marh ad Balarsha (Marh ad Balarsha 0) called a oal labelg as a E-sper erex agc oal labelg : EG q They sded he E-sper erex agcess o ee reglar graphs Mos recely Wag ad Zhag (Wag ad Zhag 04) exeded he resls od he arcle (Marh ad Balarsha 0) Theore (Swaaha ad Jeyah 00) A pah P s E-sper erex agc ad oly s odd ad Theore (Swaaha ad Jeyah 00) I a o-ral graph G o order p ad sze q s sper erex agc he he agc cosa s ge by p q q q p Theore (Marh ad Balarsha 0) Eery ree T o ee order s o E-sper erex agc I hs arcle we parally sole he ollowg ope probles ge by Marh Sgaya Kalaa ad Balarsha (Marh e al 0) Ope proble 4 Dscss he E-sper erex agcess o B whe Ope proble Fd all E-sper erex agc woded ss C e
3 06 G Marh e al Solo o he Probles Deo A broo B s deed by aachg -d peda edges wh ay oe o he peda erces d o he pah P d Theore The broo B s E-sper erex agc ad oly s odd ad Proo: As he broo B s soorphc o a pah P he resls ollow edaely ro Theore Theore The broo B s o E-sper erex agc or Proo: Le B V ad le E B The B has p erces ad q edges As B s a ree accordg o Theore ca be ee Asse ha s odd Sppose B s E-sper erex agc The here exss a E-sper erex agc oal labelg o B say Accordg o Theore he agc cosa s ge by p q q q p
4 AAM: Ier J Vol 0 Isse (Deceber 0) 07 ow we deere he salles possble ale o he s The salles possble ale o he se : s : q p Also he salles possble ales o ad are ad respecely Thereore he salles possble ale o he aboe s s: Thereore he ale o he s a s a leas exceeds a coradco Hece he resl ollows oe ha exceeds by 4 Deo 4 The s graph C s deed as ollows: } { } { } { ) ( } { ) ( C E C V The graph e C s obaed by reog } { ad he edges adace o he ro C Ths s reerred as a woded s
5 08 G Marh e al Theore A woded s Proo: C e s E-sper erex agc ad oly s odd ad Le C e be a woded s where s a pose eger sch ha The V( C e) { } { } ad E ( C e) { } { } Ths C e has p erces ad q edges C Asse ha e s E-sper erex agc The by Theore he agc cosa s ge by p qq q p p p p p p 4 whch s a eger oly whe s odd Sppose = The ad ( V C e) { } { } E ( C e) { } { } Ths C e has p erces ad q edges Sce C e s E-sper erex agc has a E-sper erex agc labelg Thereore or each erex V C e he oal wegh w (he s o he labels o he erex ad ha o he edges cde o ) s a cosa Sce here are - pede erces ad - pede edges he oly
6 AAM: Ier J Vol 0 Isse (Deceber 0) 09 possble se o he labels o he pede erces s 4 p q ad he oly possble se o he labels o he pede edges s q Thereore a he erex he larges possble label o he erex s - ad he larges possble labels o he wo edges cde o hs erex are - ad Ths he larges possble ale w s Tha s he larges possble ale o w s less ha Thereore w a coradco Hece s odd ad Coersely asse ha s odd ad The he he woded s C e I erex agc ow le Dee a oal labelg s he cycle C By Theore C e s E-sper as ollows: : V E 4 Fro he aboe labelg we hae V C e ow =
7 0 G Marh e al Thereore or eery e C V Hece s a E-sper erex agc labelg o e C V wh agc cosa where s odd ad A llsrao or Theore s ge Fgre
8 AAM: Ier J Vol 0 Isse (Deceber 0) Fgre A E-sper erex agc labelg o C 9e Coclso ad Scope I hs arcle we hae soled wo ope probles ge by Marh Sgaya Kalaa ad Balarsha (Marh e al 0) There s aoher ope proble or rher esgao he sae paper whch s ge as ollows: Ope proble Characerze all E-sper erex agc rees o odd order Acowledge The ahors wold le o express her grade o he aoyos reerees or her d sggesos ad sel coes o he orgal ascrp whch resled hs al erso REFERECES Galla J A (04) A dyac srey o graph labelg Elecro J Cob 6: #DS6 Harary F (969) Graph Theory Addso- Wesley MacDogall J A Mller M Sla Walls W D (00) Verex agc oal labelgs o graphs Ul Mah 6: - MacDogall J A Mller M Sgeg K A (004) Sper erex agc oal labelgs o graphs Proc o he 6 h Asrala Worshop o Cobaoral Algorhs pp - 9 Marh G ad Balarsha M (0) E-sper erex agc labelgs o graphs Dscree Appl Mah 60:
9 G Marh e al Marh G Sgaya B Kalaa S Balarsha M (0) E-sper erex agc labellg o graphs ad soe ope probles Appl Appl Mah 0 (): 6-4 Marr A M ad Walls W D (0) Magc Graphs Brhaser - Sprger Secod edo Boso Sedláče J Proble 7 (96) Theory o Graphs ad s Applcaos Proc Sypos pp 6-67 Swaaha V ad Jeyah P (00) Sper erex agc labellg Ida J Pre Appl Mah 4 (6): 9-99 Wag ad Zhag (04) oe o E-sper erex agc graphs Dscree Appl Mah 78: 60-6
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