Polygon Semi Designs

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1 Amercan Journal of Mathematcs and Statstcs 0, (): 4-9 DOI: 0.59/j.ajms olygon Sem Desgns Shoabu Dn, Khall Ahmad,* Deartment of Mathematcs, Unersty of the unjab Lahore Deartment of Mathematcs, (Unersty of Management and Technology Lahore, astan) Abstract In ths aer a famly of sem desgn (,, ) λ, <, λ > 0, named as olygon sem desgn s ntroduced. These desgns are generated from olygon grahs, olygon grahs are bcubc smle grahs. olygon grahs ossess Hamltonan cycles and hae grth 6. (,) olygon grah s somorhc to a famous aus grah. olygon sem desgns are symmetrc. Seeral results are roed on these desgns. Uer and lower bounds for stong sets are determned. Incdence matrx of olygon sem desgn can be used as arty chec matrx of a famous low densty arty chec codes. Keywords olygon rah, rth, Hamltonan Cycle, Bartte rah, Combnatoral Desgn, Stong Set. relmnares and Introducton A grah s a trle consstng of a ertex set V= V(), an edge set E=E() and a ma that assocates to each edge two ertces (not necessarly dstnct) called ts end onts. A loo s an edge whose end onts are equal. Multle edges are edges hang same end onts. A smle grah s one hang no loos or multle edges. To any grah, we can assocate the adjacency matrx A whch s an n n matrx (n=ii) wth rows and columns ndexed by the elements of ertex set and the (x,y)-th entry s the number of edges connectng x and y.cubc grahs also nown as tralent grahs are ntensely studed n grah theory. The bartte cubc grahs wdely nown as bcubc grahs too secal nterest. In 884,.. Tat conjectured that eery -connected lanar cubc grah has a Hamltonan cycle. Tutte, n 946, roded a counter-examle to the conjecture by constructng a 46-ertex grah. In 97, Tutte [] conjectured that all -connected bcubc grahs are Hamltonan. J.D. Horton [] n 976 roded a counter-examle to the conjecture by constructng a 96-ertex g rah. olygon grahs are structured by connectng odd number of coes of olygons of same sze n a delcate manner. These grahs are defned only for olygon wth een number of sdes and denoted by (m,), where m s the number of sdes and + s the number of coes of olygon. These grahs are bcubc wth some nterestng roertes. rncle of mathematcal nducton s used to roe the exstence of Hamltonan cycles n these grahs. It * Corresondng author: hallahmadshah@gmal.com (Khall Ahmad) ublshed onlne at htt://journal.saub.org/ajms Coyrght 0 Scentfc & Academc ublshng. All Rghts Resered s shown that these grahs hae grth 6. It s a smle fact that cubc Hamltonan grahs hae at least two Hamltonan cycles. An ncdence structure s defned on olygon smle grah (m,) through a matrx H( mt, ), where ro ws of H( mt, ) are taen as onts and columns as blocs. Let V = {,,,... }. A collect on β of dstnct subsets of V s called olygon Sem Desgn ( λ,, ) f <, λ > 0 and a) Each set n β contans exactly elements. b) Each two element subset of V s contaned n at most λ of the sets n β. The sets of β are called blocs and the number of blocs n β s denoted by b ; the set V s called the base set. Balanced Incomlete Bloc Desgns are dfferent from olygon Sem Desgns as n BIBD, Each two element subset of V s contaned n exactl y λ of the sets n β. Whereas n SD, Each two element subset of V s contaned n at most λ of the sets n β [8]. Examle. Let, m = t = + =. Then V = {,,, 4,5,6} β = {{,, },{,, 4 },{, 4,6 },{, 4,5 },{,5,6 },{,5,6} }, s a SD ( 6,, ) nduced uon It s roed that olygon sem desgns are symmetrc. We also studed stong sets n SD,s whch essentally requre combnatoral mathematcs[6]. Incdence matrx of olygon sem desgn can be used as arty chec matrx of a famous low densty arty chec

2 Amercan Journal of Mathematcs and Statstcs 0, (): codes. The sze of the smallest stong set n LDC codes hels n analyzng ther erformance under terate decodng, just as mnmum dstance hels n analyzng the erformance under maxmum lelhood decodng[7]. The structure of the aer s as follows. In secton, we ntroduce the defntons and notaton used n the later sectons; t also ncludes some results on olygon smle grahs. In secton olygon sem desgns are defned on olygon smle grahs, a lower bound for the sze of the smallest stong set n a SD s roosed.. olygon rah Let be a olygon wth m ( m ) sdes. lace +,,,,,, coes of denoted by n arallel such that s n the mddle, coes on the rght sde of say,, and coes on the left sde of,, as follows.,,,,,,. say Vertces of, and ( 0) ( 0) ( 0) are denoted by,,, m, u, u,, um w, w,, w m =,, resectely. Smle connected grah s constructed by drawng edges, such that an een ertex s connected wth odd ertex and an odd ertex wth een one n the followng manner.. Edges between ertces of dfferent olygons. a) For j< connect m ertces of j wth m ertces of j and m ertces of j wth m ertces of j +. Smlarly connect m ertces of and m ertces of j j wth m ertces of j +. wth m ertces of j b) For j = m ertces of j are already joned wth m ertces of j where the remanng m ertces of are joned wth remanng m ertces of where j =,, j. j. Edges between ertces wthn a olygon Only sdes of olygon reresent edges n Ths smle grah s denoted by ( m, ) s named as olygon grah. s somorhc to a famous bcubc symmetrc (,) dstance-regular aus grah wth 8 ertces. It has followng reresentatons. F gure. olygon grah (, ) Theorem.: Let (, ) ) V = m (+) E = m (+) m, s a cubc grah ) ) (, ) m s a bartte grah. m be a olygon grah. then. roof: ) Snce each olygon has m number of ertces and there m. are (+) coes of olygons n (, ) V = m (+) th = V E = d( ) where d ( ) ertex. m( + ) = s the degree of d ( ) = { m ( + ) } = m( + ) =, ) By defnton of m t s clear that each ertex of a olygon s connected wth two ertces of the same olygon and wth one ertex of a olygon ether on ts rght or on ts left. Hence degree of each ertex becomes three. m, s a regular smle grah of degree three.e. So (, ) m s a cubc grah. ) There exst a ertex labelng such that each een ertex s connected wth three odd ertces and each odd ertex s connected wth three een ertces, therefore ertces of can be colored usng only two colors. As eery two colorable grah s bartte. Hence s bartte grah wth equal number of ertces n each art. Theorem.: ( m, ) s a Hamltonan grah.e. t contans a Hamltonan cycle!. roof: Let m =, we use mathematcal nducton on, to roe the result. For = (,) contans Hamltonan ath as shown n fg.

3 6 Shoabu Dn et al.: olygon Sem Desgns F gure. olygon grah (,).e. Let (, ) contan Hamltonan cycle for ( 0) ( 0) ( 0) ( 0),,,, u, u, u, u u, u, u, u, w, w, w, w, 4 w, w, w, w 4 Now we roe that (, ) cycle for = + Relace edges between between, + and ( 0) contans Hamltonan and by drawng edges, +. Now we hae two + each of degree ertces of + and two ertces of two; now connect these ertces to mae the followng ath. ( 0) ( 0) ( 0) ( 0),,,, u, u, u, u ( + ) ( + ) + ( + ) u, u, u, u, u, u, u, u 4 ( + ) ( + ) ( + ) ( + ) ( ) ( ) ( ) ( ) w, w, w, w w, w, w, w,. 4 4 ( 0) w, w, w, w4 s a Hamltonan ath. Smlarly t could be roed that for arbtrary m, (, ) Theorem.: Let (, ) m has Hamltonan cycle. m be a smle grah, where m, Then the grth of ( m, ) s 6.e. ( (, )) 6. roof: Snce (, ) g m = m s a bartte grah. The grth must be an een number. We consder two cases deendng on the tyes of cycles n ( m, ) Case : Cycles contanng the ertces of one olyg on. Snce there are m ( m ) ertces n each olygon, ( ) g m, 6 Case : Cycles nol ng the ertces of more than one olyg on. In ths case the shortest cycle must nole at least two ertces from two adjacent olygons.e. two edges from each olygon. Moreoer one edge s requred to swtch from one olygon to the other and another edge to come bac. Thus ( ) g m, = 6. olygon Sem Desgn.. Defntons Let (, ) H mt where t = + be a matrx whose columns reresent odd labeled ertces and rows reresent een labeled ertces of a olygon smle grah m, or ce ersa, such that

4 Amercan Journal of Mathematcs and Statstcs 0, (): H ( mt, ) = hj = mt mt 0 Examle.. Refer to the olygon grah H (,) = f th odd ertex s ncdent wth jth een ertex (, ) n fgure Otherwse Theorem.: Let SD ( λ,, ) be a olygon Sem Desgn. Then = b..e. SD s symmetrc. roof: Fro m theorem 5. total number of ertces equals m + = mt, where t = + whch s an een number. Hence f we label all ertces from,,...mt, snce een label ertces are taen as onts.e. = mt and odd ertces as blocs b = mt, Hence = b λ,, s a olygon sem desgn Theorem. If SD then each element of the base set occurs n r blocs, where b = r and = r roof: To roe the clam we count n two ways the cardnalty of the set. {(, ), } X= xb x V B β for each x V the bloc B can be chosen n r dfferent ways, hence by roduct rule X r; other hand, for each of the b blocs x can be chosen n dfferent ways, agan by roduct rule X = b, So b = r Snce = b = r.. Incdence Matr x If SD s a ( λ,, ) bnary b = on the olygon sem desgn then the matrx A ( a j ) = where f th bloc of β contans jth ont of. aj = 0 other wse Is called ncdence matrx of the SD. Of course such a matrx s by no means unque, but deends on the order n whch we wrte the blocs and onts. By defnton, each column contans contan, s and accordng to the theorem.. each row also, r, s = condton n defnton means that f we c any two columns there are at most λ rows n whch there s a n both these columns. For m any two rows(columns) of (, ) H mt hae at most one n common.e. n the same column(row). So λ = Although by theorem.. SD s symmetrc, t does not mean that ts ncdence matrx should be symmetrc. λ not just In an ncdence matrx A of the SD (,, ) each row but also each column has exactly ' s and not just eery two rows but also eery two columns hae at most λ 's n common therefore also A t s an ncdence matrx for so me ( λ,, ) SD, Also notce that by fsher s n equalty, the transose of an ncdence matrx desgn could be an ncdence matrx of a desgn only f b=... Stong Sets Defnton: A set S { B, B,... Bs} olygon sem desgn ( λ,, ) UB, s = { } = of blocs n a s called stong set f S = B S B s of order at least. Let s mn be the sze of smallest non emty stong set.... Bounds For Stong Sets Theorem.. Let Dm (,,) m be a olygon symmetrc sem desgn. If S s a stong set n D then. smn = S 4.

5 8 Shoabu Dn et al.: olygon Sem Desgns roof As S be a stong set such that s S = and B = m.e. number of onts s all blocs of S. B S Count the set {,, } X= B B S B n two dfferent ways. Snce there are m onts n B and B S each ont les n at least two blocs B S by defnton of s at least S. So the total number of such ars ( B, ) m. On the other hand there are s blocs n S and each bloc contanng three onts so there we exactly s such ars exst. s m Now by ncluson excluson rncle m s B B j s m s B, Bj S B, Bj S j s B Bj, j Any two blocs ntersect n most two onts. s s s s 6s s s s s s s s 4 λ olygon sem desgn, for V Let (,, ) defne that as the collecton of blocs B also { B β B} = Theorem.. Let (,,) wth set. B we β such s the set β \, be a olygon sem desgn, s a stong r = then = { B β B} roof: Let most bloc of V, for each, should be n at,[by defnton of olygon sem desgn any ar of onts should be contaned by at most one bloc]. Sne each V les n exact ly r =, blocs n SD, therefore each ont n blocs n. So must be ncdent wth at least s a stong set. Suose by contradcton for some two blocs of les n, and les n more three blocs of. So there s a ar of onts and, whch les n more than one blocs of SD, ths contradcts the fact that each ar of onts les n at most one bloc. Hence each ont n. can be ncdent n at most one bloc n Corollary.. In (,,) olygon Sem Desgn Smn b roof: For each V, = b = β = + = b Smn b 4. Conclusons olygon sm le grah s a new dscrete structure. We wored out for some basc roertes. olygon grahs are bartte, therefore could be used as Tanner grahs to generate low densty arty chec codes. A olygon sem desgn s agan a recent deeloment n combnatoral mathematcs. Some results are roed on these desgns. The bounds roed for stong sets, are to be used as a erformance ndcator for the LDC codes defned on olygon sem desgns, these codes are wdely used for error deducton and correcton. REFERENCES [] Tutte, W. T. "On Hamltonan Crcuts." J. London Math. Soc., 98-0, 946 [] Horton, J. D. "On Two-Factors of Bartte Regular rahs." Dsc. Math. 4, 5-4, 98. [] J Read, R. C. and Wlson, R. J. An Atlas of rahs. Oxford, England: Oxford Unersty ress, 998 [4] Renhard Destel. rah Theory. Electronc Edton 000. c Srnger-Verlag New Yor 997, 000

6 Amercan Journal of Mathematcs and Statstcs 0, (): [5] C. Bazgan, M. Santha, Z. Tuza: "On the Aroxmaton of Fndng A(nother) Hamlton Cycle n Cubc Hamlton rahs" (Extended Abstract). STACS 998: [6] J.H. an Lnt and R.M. Wlson, "A Course n Combnatorcs, nd ed.", Cambrdge Unersty ress,00. Internatonal Symosum on Volume, Issue, 9 June-4 July 00 [8] Charles J. Colbourn and Jeffrey H. Dntz. " Handboo of combnatoral desgns." Chaman & Hall/CRC, Boca Raton, FL, 007 [7] Kashya, N.; Vardy, A. Stong Sets n Codes from Desgns Informaton Theory, 00. roceedngs. IEEE