On the Existence of n-tuple Magic Rectangles

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1 Proc. of he Third Il. Cof. o Adaces i Alied Sciece ad Eiromeal Egieerig ASEE 0 Coyrigh Isie of Research Egieers ad Docors, USA.All righs resered. ISBN: 90 doi: 0./ 900 O he Exisece of Tle Magic Recagles Phaisacha Iooai ad Thirade Jiarasksak Absrac Magic recagles are a classical geeralizaio of he wellkow magic sares, ad hey are relaed o grahs. A grah G is called degreemagic if here is a labellig of he edges by iegers,,..., EG ( ) sch ha he sm of he labels of he edges icide wih ay erex is eal o ( E( G) )deg( ) /. I his aer we geeralize magic recagles o be le magic recagles, ad roe he ecessary ad sfficie codiios for he exisece of ee le magic recagles. Usig his exisece we ideify he sfficie codiio for degreemagic labelligs of he fold selfio of comlee biarie grahs o exis. Keywords magic sares, magic recagles, degreemagic grahs I. Irodcio Magic recagles are a aral geeralizaio of he magic sares which hae widely iriged mahemaicias ad he geeral blic. A magic (, ) recagle R is a array i which he firs osiie iegers are laced sch ha he sm oer each row of R is cosa ad he sm oer each colm of R is aoher (differe if ) cosa. Harmh [, ] sdied magic recagles oer a cery ago ad roed ha Theorem ([, ]) For,, here is a magic (, ) recagle R if ad oly if (mod ) ad (, ) (, ). I 990, S [] sdied he exisece of magic recagles. Laer, Bier ad Rogers [] sdied balaced magic recagles, ad Bier ad Kleischmid [] sdied cerally symmeric ad magic recagles. The Hagedor [] reseed a simlified moder roof of he ecessary ad sfficie codiios for a magic recagle o exis. The coce of magic recagles was geeralized o dimesios ad seeral exisece heorems were roe by Hagedor []. For simle grahs wiho isolaed erices, if G is a grah, he VG ( ) ad EG ( ) sad for he erex se ad he edge se of G, reseciely. Cardialiies of hese ses are called he order ad size of G. Le a grah G ad a maig f from EG ( ) io osiie iegers be gie. The idex maig of f is he maig f from VG ( ) io osiie iegers defied by Phaisacha Iooai ad Thirade Jiarasksak Dearme of Mahemaics, Facly of Sciece, Kig Mogk s Uiersiy of Techology Thobri Pracha Uhi Rd., Bag Mod, Thg Khr, Bagkok 00, Thailad f ( ) (, e) f ( e) for eery V( G), () ee ( G) where (, e) is eal o whe e is a edge icide wih a erex, ad 0 oherwise. A iecie maig f from EG ( ) io osiie iegers is called a magic labellig of G for a idex if is idex maig f saisfies f () for all V( G). () A magic labellig f of a grah G is called a sermagic labellig if he se f ( e) : e E( G) cosiss of cosecie osiie iegers. A grah G is sermagic (magic) wheeer a sermagic (magic) labellig of G exiss. A biecie maig f from EG ( ) io {,,..., EG ( ) } is called a degreemagic labellig (or oly dmagic labellig) of a grah G if is idex maig f saisfies EG ( ) f ( ) deg( ) for all V( G). () A dmagic labellig f of G is called balaced if for all V( G), he followig eaio is saisfied e E( G) : (, e), f ( e) E( G) / e E( G) : (, e), f ( e) E( G) /. A grah G is degreemagic (balaced degreemagic) or oly dmagic whe a dmagic (balaced dmagic) labellig of G exiss. The coce of magic grahs was irodced by Sedláček []. Laer, sermagic grahs were irodced by Sewar [9]. There are ow may aers blished o magic ad sermagic grahs; we refer he reader o Gallia [0] for more comrehesie refereces. Recely, he coce of degreemagic grahs was irodced by Bezegoá ad Iačo [] as a exesio of sermagic reglar grahs. They also esablished he basic roeries of degreemagic grahs ad roed ha Proosiio ([]) For,, he comlee biarie grah K, is dmagic if ad oly if (mod ) ad (, ) (, ). Theorem ([]) The comlee biarie grah K, is balaced dmagic if ad oly if he followig saemes hold: (i) 0 (mod ); (ii) if (mod ), he mi{, }. I his aer we irodce le magic recagles. To show heir exisece, we irodce he closely relaed coce ()

2 Proc. of he Third Il. Cof. o Adaces i Alied Sciece ad Eiromeal Egieerig ASEE 0 Coyrigh Isie of Research Egieers ad Docors, USA.All righs resered. ISBN: 90 doi: 0./ 900 of cerally le symmeric recagles. The we se he exisece of cerally le symmeric recagles o gie a cosrcio of ee le magic recagles. Fially, we ideify he sfficie codiio for dmagic labelligs of he fold selfio of comlee biarie grahs o exis. II. The Tle Magic Recagles I his secio we irodce le magic recagles ad roe he ecessary ad sfficie codiios for ee le magic recagles o exis. Defiiio A le magic (, ) recagle R r i, : ( ) ( r )...( r ) is a class of arrays i which each array has rows ad colms, ad he firs osiie iegers are laced sch ha he sm oer each row of ay array of R is cosa ad he sm oer each colm of R is aoher (differe if ) cosa. Le R : ( ri, )( ri, )...( ri, ) be a le magic (, ) recagle. As each row sm of ay array of R is ( ) / ad each colm sm of R is ( ) / ad boh are ieger, we he hae Proosiio If R is a le magic (, ) recagle, he he followig saemes hold: (i) if is odd, he (mod ); (ii) if is ee, he 0 (mod ). Proosiio allows he se of le magic recagles o be diided io ses of odd ad ee recagles. We ickly see ha a le magic (, ) recagle does o exis. To show he exisece of oher ee le magic recagles, we irodce he closely relaed coce of cerally le symmeric (, ) recagles as follows. Defiiio Le x ad le R be a class of ee recaglar arrays i which each array has rows ad colms ad he eries of R are mbers ( x ),..., ( x / ). R is a cerally le symmeric (, ) recagle of ye x if he sm oer each row ad colm of ay array is zero. Addiioally, if R has a eal mber of osiie ad egaie mbers i each row ad colm of ay array, we say ha R is balaced. If R is a ee le magic (, ) recagle, he by sbracig ( ) / from each ery of R, we obai a cerally le symmeric (, ) recagle of ye /. Similarly, eery cerally le symmeric (, ) recagle of ye / deermies a ee le magic (, ) recagle. Ths, we ca se he exisece of cerally le symmeric (, ) recagles o roe he exisece of ee le magic (, ) recagles. Lemma For x, y, if a balaced cerally le symmeric (, ) recagle of ye x exiss, he a balaced cerally le symmeric (, ) recagle of ye y exiss. Proof. Sose ha R : ( r )( r )...( r ) is he gie recagle. The we defie a (, ) recagle...( s ) by S : ( s )( s ) s ( y x)sg( r ) r, for eery {,,..., }. The eries of S are he mbers ( y ),..., ( y / ). For ay {,,..., } ad i, he sm of each row is s ( y x)sg( ri, ) ri, ( y x) sg( r ) r 0, ad for all, he sm of each colm is s ( y x)sg( ri, ) ri, i i i i ( y x) sg( r ) r 0. Ths, S is a cerally le symmeric (, ) recagle of ye y. For ay {,,..., }, if r is osiie, he r x m for some m. Hece, s y m is also osiie. Similarly, r egaie imlies s egaie. Therefore, S is balaced. Proosiio If a balaced cerally le symmeric (, ) recagle exiss, he a le magic (, ) recagle exiss. Proof. Sose R is he gie recagle. If R has ye x, he by Lemma, here exiss a balaced cerally le symmeric (, ) recagle of ye /. Therefore, a le magic (, ) recagle exiss. Examle We cosider a balaced cerally le symmeric (, ) recagle R : ( ri, )( ri, )...( ri, ) of ye as follows R : The we defie a le (, ) recagle...( s ) relaed o R by S : ( s )( s ), sg(, ) si ri ri,, for eery {,,,, }.

3 Proc. of he Third Il. Cof. o Adaces i Alied Sciece ad Eiromeal Egieerig ASEE 0 Coyrigh Isie of Research Egieers ad Docors, USA.All righs resered. ISBN: 90 doi: 0./ 900 Ths, S is a balaced cerally le symmeric (, ) recagle S of ye / as follows S : By addig / o each ery of S, we obai a le magic (, ) recagle T as below T : Clearly, he sm oer each row of ay array is ad he sm oer each colm is. Proosiio If a balaced cerally le symmeric (, ) recagle R ad a cerally le symmeric (, ) recagle S exis, he a cerally le symmeric (, ) recagle T exiss. If S is a balaced recagle, he T ca also be chose o be balaced. Proof. Sose S has ye x. By Lemma, we kow ha here exiss a balaced cerally le symmeric (, ) recagle R of ye x /. The by sackig R ad S ogeher, we obai a recagle T whose rows ad colms sm is zero. Ths, T is a cerally le symmeric (, ) recagle of ye x. If S is balaced, he i is easy o see ha T is also balaced. Sice le magic (, ) recagles corresod o cerally le symmeric (, ) recagles of ye /, we hae he followig corollary. Corollary Sose a le magic (, ) recagle ad a balaced cerally le symmeric (, ) recagle exis. The a le magic (, ) recagle exiss. Usig he coce of a cerally le symmeric recagle, we ca roe he exisece of ee le magic recagles. Or ools are he balaced cerally le symmeric (, ) recagle A: ( a )( a )...( a ) gie by ( a ), ad he le magic (, ) recagle ( b ) gie by B : ( b )( b )... i, i, ad b, b, for all {,,..., }. ( ) if, ( ) if, ( ) if, 9 ( ) if, ( ) if, ( ) if, ( ) if, ( ) if, 0 ( ) if, ( ) if, ( ) if, ( ) if, Proosiio Le be a ee ieger. The a le magic (, ) recagle exiss. Proof. We idc o. The exisece of le recagles A ad B shows ha we eed oly roe he roosiio for. Assme we kow ha a le magic (, ) recagle exiss for all ee. The we kow a le magic (, ) recagle R exiss. By Corollary, we ca add R ad A ogeher o form a le magic (, ) recagle. Proosiio Le ad be ee osiie iegers wih (, ) (, ). The a le magic (, ) recagle exiss. Proof. By Proosiio, we ca assme ha. Usig A ad Proosiio, idcio shows ha a balaced cerally le symmeric (, ) recagle R exiss. Ths, a le magic (, ) recagle exiss ad we ca assme ha. Now assme ha a le magic (, ) recagle exiss for all ee. We he kow ha a le magic (, ) recagle S exiss. By Corollary, we ca add R ad S ogeher o gie a le magic (, ) recagle. Examle The followig arrays are examles of ee le magic recagles. A rile magic (, ) recagle

4 Proc. of he Third Il. Cof. o Adaces i Alied Sciece ad Eiromeal Egieerig ASEE 0 Coyrigh Isie of Research Egieers ad Docors, USA.All righs resered. ISBN: 90 doi: 0./ 900 The each row sm of ay array is ad each colm sm of ay array is 9. A le magic (, ) recagle The each row sm ad each colm sm of ay array i a recagle eals 0. The Fold SelfUio of Comlee Biarie Grahs III. For ay ieger, he fold selfio of a grah G, deoed by G, is he io of disoi coies of G. I his secio we ideify he sfficie codiio for degreemagic labelligs of he fold selfio of comlee biarie grahs K K K... K o exis.,,,, Theorem For ay ieger ad ee iegers,, h le K, be he coy of K, for all {,,..., }. A maig f from E( K, ) io osiie iegers gie by f ( i ) ri, for eery i E( K, ), is a dmagic labellig of K, if ad oly if...( r ) is a le magic (, ) recagle. R : ( r )( r ) i, i, Proof. Le U {,,..., } ad V {,,..., } be arie ses of K,. Sose ha R is a le magic (, ) recagle. The f is a biecio from E( K, ) oo {,,..., }. For ay U, we hae i ri, f i f i For all z, we hae By (), we hae By (), we hae ( ) ( ) s s s, f ( ) f ( ) r. ri, f i f i i ( ) ( ) z i z z i i f ( ) f ( ) r. ri, rs, ri, ri, z i i ( ). ( ). Therefore, R is a le magic (, ) recagle. Accordig o Theorem ad Proosiio, we obai he followig resl. Proosiio Le ad be ee osiie iegers wih (, ) (, ). The K, is a dmagic grah for all iegers. Examle We ca cosrc a dmagic grah K, (see Figre ) wih he labels o edges i, ad, i TABLE I. i of, () () K, where i i f ( ) f ( ) r ad for ay V, we hae ( ) deg( i ), i i i f ( ) f ( ) r ( ) deg( ). i.e., f is a dmagic labellig of K,. Figre. A dmagic grah K,. Now sose ha f is a dmagic labellig of K,. For all i s, we hae

5 Proc. of he Third Il. Cof. o Adaces i Alied Sciece ad Eiromeal Egieerig ASEE 0 Coyrigh Isie of Research Egieers ad Docors, USA.All righs resered. ISBN: 90 doi: 0./ 900 TABLE I. THE LABELS ON EDGES OF DMAGIC GRAPH K, Verices [0] J.A. Gallia, A dyamic srey of grah labelig, Elecro. J. Combi., #DS, ol., 009. [] L. Bezegoá ad J. Iačo, A exesio of reglar sermagic grahs, Discree Mah., ol. 0,., Verices Verices Ackowledgmes This work was sored by Raamagala Uiersiy of Techology Laa ad Dearme of Mahemaics, Facly of Sciece, Kig Mogk's Uiersiy of Techology Thobr Thailad. Refereces [] T. Harmh, Über magische Qadrae d ähiche Zahlefigre, Arch. Mah. Phys., ol.,.,. [] T. Harmh, Über magische Rechecke mi gerade Seiezahle, Arch. Mah. Phys., ol.,.,. [] R. S, Exisece of magic recagles, Nei Mogol Daxe Xebao Zira Kexe, ol., o.,. 0, 990. [] T. Bier ad G. Rogers, Balaced magic recagles, Eroea J. Combi., ol.,. 99, 99. [] T. Bier ad A. Kleischmid, Cerally symmeric ad magic recagles, Discree Mah., ol.,. 9, 99. [] T. Hagedor, Magic recagles reisied, Discree Mah., ol. 0,., 999. [] T. Hagedor, O he exisece of magic dimesioal recagles, Discree Mah., ol. 0,., 999. [] J. Sedláček, Theory of grahs ad is alicaios, Proc. Sym. Smoleice, Problem, Praha,., 9. [9] B.M. Sewar, Magic grahs, Caad. J. Mah., Vol.,. 009, 9.