sthanik resheey aalekh

aalekh siddhantat, sthanik resheey aalekh haah eka dishaheen aalekh aahe jyamadhye pratyek kadaa agadi ekach trikonachi asate. samaan ritine, aalekhatil pratyek shirshasathi, tyache shejari heey pratyekjan agadi eka bindunna shejarila lagoon asatat, tyamule shejarila prerit julanimadhye jodale jaau shakate. sthanik resheey aalekhanna sthanik patalivar julanare aalekh dekhil mhatale gele aahet. tyanche trikon trikon-mukt 3-eksaman rekheey hypergraphche atiuchchare aanee kahee aanshik steiner tiheri pranalinche khand tayar karatat aanee sthanik rekheey aalekh heey yaa hypergraph kinwa aanshik steiner pranalinche achookpane gefmann aalekh aahet.
sthanik resheey aalekhasathi anek rachana dnyat aahet. sthanik resheey aalekhanchya udaharananmadhye trikoni cactus aalekh, 3-niyamit trikon-mukt aalekhanche resha aalekh aanee sthanik patalivar lahan resheey aalekhanche cartasian utpadane yancha samavesh hoto. kahee gudghyache aalekh aanee kahee jordar niyamit aalekh dekhil sthanik patalivar resheey asatat.
sthanik patalivar resheey aalekhanchya kitee kadaa asoo shakatat haah prashna ruza-szemereddy samasyechya sutranpaiki eka aahe. jaree ghan aalekhamadhye shirobinduchya sankhyechya varganchya pramanat anek kadaa asoo shakatat, taree sthanik resheey aalekhamadhye kadaa qamee asatat, jyaa kiman eka chhotya sthir nasalelya ghatakane chaurasapeksha qamee padatat. sthanik patalivar resheey asoo shakanare sarvat daat samtol aalekh dekhil dnyat aahet. trikoni cactus aalekh heey sthanik patalivar sarvat qamee daat resheey aalekh aahet.
bandhkam.
gluing aanee utpadane.
maitriche aalekh, ekach samayik shirshasthani trikonanchya sangrahala ekatra chikatavun tayar kelele aalekh, sthanik resheey aahet. teey ekmev maryadit aalekh aahet jyancha adhik majboot gunadharm aahe qee shirobinduchya pratyek jodila (shejari kinwa nahee) agadi eka samaan shejari samayik karatat. adhik samanyatah pratyek trikoni cactus aalekh, joou konatyahi atirikt chakranchi nirmiti naan karata samayik shirobinduvar trikonanna chikatavun tayar kelela aalekh sthanik resheey asato.
sthanik resheey aalekh heey khaleel krutidvare sthanik patalivarchya lahan resheey aalekhatun tayar kele jaau shakatat, jeey aalekhavaril samooh-berries prakriyecha eka prakar aahe.
sootra _ 1 aanee sootra _ 2 heey konatehi done sthanik resheey aalekh asoo dya, tyapaiki pratyekatun eka trikon nivada aanee done nivadalelya trikonanmadhye shirobinduchya sambandhit jodya ekatra misaloon done aalekh chikatwa. tyanantar parinami aalekh sthanik patalivar resheey rahato.
konatyahi done sthanik resheey aalekhanche cartasian utpadan sthanik patalivar resheey rahate, kaaran utpadanatil konatehi trikon eka kinwa itar ghatakanchya trikonatun yetat. udaharanarth, nauu-vertex pally aalekh (3-3 dvidhruviyatecha aalekh) haah done trikonancha cartasian gunakar aahe. hamming aalekh sootra _ 3 heey sootra _ 4 trikonanche cartasian utpadan aahe aanee punha sthanik resheey aahe.
lahan aalekhatun.
kahee aalekh jeey swatah sthanik resheey nasatat, tyancha vapar sthanik patalivar mothe resheey aalekh tayar karnyasathi eka chaukat mhanoon keela jaau shakato. asha eka rachnet resha aalekh samavisht aahet. konatyahi aalekh sootra _ 1 sathee, resha aalekh sootra _ 6 haah eka aalekh aahe jyamadhye sootra _ 1 chya pratyek kathasathi eka shirobindu asato. sootra _ 6 madheel done shirobindu jevha sootra _ 1 madhye darshavilelya done kadaa samaan antim bindu asatat tevha shejari asatat. jarr sootra _ 1 haah 3-niyamit trikon-mukt aalekh asel, tarr tyache resha aalekh sootra _ 6 heey 4-niyamit aanee sthanik resheey aahe. sootra _ 1 chya pratyek shirobindu sutrasathi _ 12 haah trikon aahe, jyache trikon shirobindu sootra _ 12 chya teen kadashi sambandhit aahet. pratyek 4-niyamit sthanik resheey aalekh asha prakare tayar keela jaau shakato. udaharanarth, cuboctahedroncha aalekh haah ghancha resha aalekh aahe, mhanoon too sthanik patalivar resheey aahe. cartasian utpadan mhanoon varr bandhalela sthanik resheey nauu-vertex pally aalekh, upayuktata aalekh sootra _ 15 chya reshechya aalekhapramane vegalya prakare dekhil tayar keela jaau shakato. pieterson aalekhcha resha aalekh dekhil yaa bandhkamadware sthanik resheey aahe. tyacha pinjaryansarkha gunadharm aaheh haah sarvat lahan sambhavya aalekh aahe jyamadhye sarvat mothya toliche teen shirobindu asatat, pratyek shirobindu agadi done kaath-vibhajak tolimadhye asato aanee vegvegalya tolichya kadaa asalelya sarvat lahan chakrala paach lambi asate.
adhik guntaguntichi vistar prakriya samtol aalekhanna lagoo hote. sootra _ 1 haah samatolat asha prakare antarbhut kelela samataliy aalekh asoo dya qee pratyek chehara eka chaturbhuj asel, jase qee ghancha aalekh. sootra _ 1 chya pratyek prushthbhagavar chauras pratidrupata chikatavun, aanee nantar sootra _ 1 chya mool kadaa kadhoon takalyane, eka naveen sthanik resheey samtol aalekh tayar hoto. parinamachya kadaa aanee shirobinduchi sankhya yoolarchya bahu-vibhajak sutravarun mojali jaau shakateh jarr sootra _ 1 madhye sootra _ 20 shirobindu asateel, tarr tyala nemke sootra _ 21 chehare aahet aanee sootra _ 1 chya shirobindula pratiprisamdware badalanyachya parinamanmadhye sootra _ 23 shirobindu aanee sootra _ 24 kadaa aahet. udaharanarth, 4-chakrachya done bajunmadhun (aateel aanee bahereel) asha prakare cuboctahedron punha tayar keela jaau shakato.
yaa bandhkamache kadhoon takalele 4-chakr heey cuboctahedronver tyachya chauras chauras chaar karnanche chakr mhanoon pahile jaau shakate, jeey polyhedronla dwibhajit karate.
bijganitiy rachana.
summ-aakar sanchanchya chhedanbindu namunyanpasun tayar kelele kahee gudghyache aalekh, sthanik patalivar resheey asatat. gudghyachya aalekhanche varnan done mapdandandware kele jaate, teey pratinidhitva karatat tyaa sanchancha aakar aanee vishvacha aakar jyachyapasun heey sanch kadhale jaatat. kneeser graph sootra _ 25 madhye sootra _ 26 shirobindu aahet (dwipadi gunankansathi pramanit sanketanmadhye), jeey sootra _ 28-ghatak sanchatil sootra _ 27-ghatak upasanch darshavtat. yaa aalekhamadhye, jevha sambandhit upasanch heey parasparsambandhit sanch asatat, jyat konatehi ghatak samaan nasatat, tevha done shirobindu shejari asatat. vishesh prakaranat jevha sootra _ 29, parinami aalekh sthanik resheey asato, kaaran pratyek done visangati sootra _ 27-ghatak upasanch sootra _ 31 aanee sootra _ 32 sathee tyaa doghanpasun nemke eka sootra _ 27-ghatak upasamuchchay visangati asate, jyamadhye serve ghatak asatat jeey sootra _ 31 kinwa sootra _ 32 madhye nasatat. parinami sthanik resheey aalekhamadhye sootra _ 36 shirobindu aanee sootra _ 37 kadaa asatat. udaharanarth, sootra 38 sathee, gudghyacha aalekh sootra 39 heey 15 shirobindu aanee 45 kadaa asalele sthanik resheey aahe.
sthanik resheey aalekh heey sankhyanchya pragati-mukt sanchatoon dekhil tayar kele jaau shakatat.
sootra _ 40 hii eka abhajya sankhya asoo dya aanee sootra _ 41 haah sankhya modulo sootra _ 40 chaa upasanch asoo dya, jenekarun sootra _ 41 chey konatehi teen sadasya ankganitiy pragati modulo sootra _ 40 tayar karoo shakanar naheet. (mhanje, sootra _ 41 heey eka selem-spensor sanch modulo sootra _ 40 aahe.) yaa sanchancha vapar formula _ 47 shirobindu aanee formula _ 48 kadaa asalela tripakshiy aalekh tayar karnyasathi keela jaau shakato joou sthanik resheey aahe. haah aalekh tayar karnyasathi, shirobinduche teen sanch banwa, pratyek sootra 49 teey sootra 50 paryant kramankit. sootra 49 teey sootra 50 paryantchya shrenitil pratyek sankhya sootra 51 sathee aanee sootra 41 chya pratyek ghatak sootra 28 sathee, shirobinduchya pahilya sanchatil sankhya sootra 51 saha shirobindula jodanara trikon tayar qaraa, shirobinduchya bindunna sanchatil sankhya sootra 57 saha shirobindu aanee shirobinduchya shirobinduchya sanchatil sankhya sootra 58 saha shirobindu tayar qaraa. yaa serve trikonancha sangam mhanoon aalekh tayar qaraa. kaaran teey trikonanche millen aahe, parinami aalekhachi pratyek dhaar eka trikonachi aahe. tathapi, asha prakare tayar jhalelya trikonanvyatirikt itar konatehi trikon asoo shakat naheet. itar konatyahi trikonala shirobindu kramank asalele sootra 59 asel, jethe sootra 28, sootra 27 aanee sootra 62 heey serve sootra sootra sambandhit aahet, jeey sootra 41 madhye konatehi ankganitiy pragati sootra 64 nahee yaa gruhitkala bhang karatat. udaharanarth, sootra 66 aanee sootra 67 saha, yaa rachanecha parinam nauu-vertex pally aalekh aahe.
sthanik resheey aalekhatil trikonanna 3-samaan hypergraph tayar karnare manale jaau shakate. assa hypergraph resheey asane aavashyak aahe, mhanje tyachya done hyperages (trikon) ekapeksha jaast shirobindu samayik karoo shakat naheet. sthanik resheey aalekh swatach hypergraphcha gefmann aalekh aahe, joou eka samanya hyperageshy sambandhit asalelya shirobindunchya jodyancha aalekh aahe. yaa drushtikonatun hypergraphchya gheryabaddal bolane arthpurn aahe. aalekhanusar, hii sarvat lahan chakrachi lambi aahe jee aalekhatil trikonanpaiki eka nahee. dhruviyata aalekhavar aadharit bijganitiy rachana (jyala tapkiri aalekh dekhil mhanatat) yaa sandarbhat, 4-chakr nasalele ghandat sthanik resheey aalekh shodhanyasathi vaparali gelee aahe; tyancha hypergraph gher paach aahe. dhruviyata aalekh haah maryadit prakshepak pratalavarun paribhashit keela jaato aanee dhruviyata mhanje tyache bindu aanee tyachya reshanmadhil ghatana-sanrakshn dwibhajan. dhruviyata aalekhatil shirobindu heey bindu asatat aanee jevha jevha eka dhruviy asato tevha eka dhaar done bindunna bindunna asalelya resheshi jodate. adhik bijganitiyadrushtya, ekach aalekhatil shirobindu eksandh nirdeshankadware darshavile jaau shakatatah heey serve shoonya navhe tarr maryadit kshetratun sootra _ 68 mulyanche tippat aahet, jethe done tippat jevha jevha ekmekanche scalar gunakar asatat tevha samatalatil samaan bindu paribhashit karatat. asha prakare tiheri gunakarane darshavilele done bindu, jevha tyanche antargat utpadan shoonya asate tevha teey shejari asatat. visham kram sootra _ 69 chya maryadit kshetrasathi dhruviyata aalekhamadhye sootra _ 70 shirobindu aahet, jyapaiki sootra _ 71 heey swatashi sanlagn aahet aanee konatyahi trikonashi sambandhit naheet. jevha heey kadhale jaatat, tevha parinam haah sthanik resheey aalekh asato jyamadhye sootra _ 72 shirobindu, sootra _ 73 kadaa aanee hypergraph gher paach asatat, jeey yaa gheryachya sthanik resheey aalekhasathi khalchya kramaparyantchya sandnyansathi kadanchi jastit jaast shakya sankhya detat.
niyamitata.
kahee shirobindu asalele niyamit aalekh.
jevha tyachya serve shirobindu samaan degree asatat, mhanje aapatkalin kadaa sankhya, tevha aalekh niyamit asato. pratyek sthanik resheey aalekhala pratyek shirshasthani summ degree asane aavashyak aahe, kaaran pratyek shirshavaril kadaa trikonat jodalya jaau shakatat. done sthanik resheey niyamit aalekhanche cartasian utpadan punha sthanik resheey aanee niyamit aahe, jyachi degree ghatakanchya anshanchya berries itaki aahe. tyamule, pratyek summ degreeche niyamit sthanik resheey aalekh tayar karnyasathi, sthanik patalivarchya done anshanchya (trikon) resheey aalekhanchi cartasian utpadane gheta yetat.
formula _ 74-niyamit sthanik resheey aalekhamadhye kiman formula _ 75 shirobindu asane aavashyak aahe, kaaran konatyahi trikonamadhye aanee tyachya shejari ektya itakya shirobindu asatat. (trikonache konatehi done shirobindu sthanik reshatmakateche ullanghan kelyashivay shejari samayik karoo shakat naheet.) agadi itakya shirobindu asalele niyamit aalekh tevhach shakya hotat jevha sootra _ 76 heey 1,2,3 kinwa 5 asel aanee yaa chaar prakarananpaiki pratyekasathi teey advitiyapane paribhashit kele jaatat. shirobindunchya sankhyevar heey bandhalele chaar niyamit aalekh mhanje 3-vertex 2-niyamit trikon sootra _ 77,9-vertex 4-niyamit pally aalekh, 15-vertex 6-niyamit gudghyacha aalekh sootra _ 39 aanee 27-vertex 10-niyamit purak aalekh. antim 27-vertex 10-niyamit aalekh dekhil ghan prushthbhagavaril 27 reshanchya chhedanbindu aalekh darshavito.
jordar niyamit aalekh.
eka majboot niyamit aalekh heey chaar gunank sootra 79 dware darshavile jaau shakate, jethe sootra 20 hii shirobinduchi sankhya aahe, sootra 81 hii prati shirobindu aaptit kadaa sankhya aahe, sootra 82 hii shirobinduchya pratyek sanlagn jodisathi samayik shejari sankhya aahe aanee sootra 83 hii shirobinduchya pratyek sanlagn nasalelya jodisathi samayik shejari sankhya aahe. jevha sootra _ 84 asate tevha aalekh sthanik patalivar resheey asato. varr aadhich namood kelele sthanik resheey aalekh jeey jordar niyamit aalekh aahet aanee tyanche mapdand aahet
itar sthanik resheey jordar niyamit aalekh samavisht aahet
sootra _ 84 saha itar sambhavya-vaidh sanyojananmadhye (99,14,1,2) aanee (115,18,1,3) samavisht aahet parantu tyaa mapdandansah jordar niyamit aalekh astitvat aahet qee nahee heey adnyat aahe. mapdandansah (99,14,1,2) majboot niyamit aalekh astitvat asanyacha prashna convechi 99-aalekh samasya mhanoon olakhala jaato aanee johnn horton convene tyachya upayasathi $1000 chey bakshis dile aahe.
antar-niyamit aalekh.
4 kinwa 6 vya anshanche maryadit antar-niyamit aalekh aahet jeey sthanik patalivar resheey aahet. samaan anshanchya majboot niyamit aalekhanchya palikade, tyamadhye pieterson aalekh, hamming aalekh sootra _ 87 aanee ardha palak aalekh yancha dekhil samavesh hoto.
ghanata.
ruza-szemereddy samasyeche eka sootra sthanik resheey aalekhamadhye _ 20-vertex sutramadhye kadaa jastit jaast sankhya vicharte. imare zed. ruza aanee endrey zemereddy yanni siddha kele qee hii kamal sankhya sootra 89 aahe parantu pratyek sootra 91 sathee sootra 90 aahe. pragati-mukt sanchanmadhun sthanik resheey aalekh tayar kelyane, sootra 92 kadansah, sthanik patalivar sarvat jaast dnyat resheey aalekh tayar hotat. (yaa sutranmadhye, sootra _ 93, sootra _ 94 aanee sootra _ 95 hii anukrame lahan oo sanketan, mothe omega sanketan aanee mothe oo sanketan yanchi udaharane aahet.)
samtol aalekhamadhye, sootra _ 20 shirobindu asalelya sthanik resheey aalekhatil kadachi kamal sankhya sootra _ 97 aahe. sootra _ 100 sathee, sootra _ 100 sathee, sootra _ 100 chya chaturbhuj chaturbhuj antiprecemedhye vistar karoon tayar kelelya sootra _ 98 shirobindu aanee sootra _ 99 kathansah bahu-hedral aalekhanchya anant kramamadhye cuboctahedroncha aalekh pahila aahe. hii udaharane darshavtat qee _ 97 heey sootra varchya maryadeparyant pohochavata yete.
pratyek sthanik resheey aalekhamadhye assa gunadharm asato qee tyatoon konatihi julani kadhoon taklyanantar too jodalela rahato, kaaran aalekhatil konatyahi margavar, pratyek julanari kadaa tyachya trikonachya itar done kathanni badalali jaau shakate. yaa gunadharmansah aalekhanmadhye, trikoni cactus aalekhanmadhye sarvat qamee datata aahe, jeey sthanik patalivar sarvat qamee daat resheey aalekhanmadhye dekhil aahet.
anuprayog.
sthanik resheey aalekhancha eka vapar greachy aakruti tayar karnyat hoto, jyacha vapar quantum logicmadhye vishisht hilbert space samikarane ekmekankadun anuman lavali jaau shakatat qee nahee heey nirdharit karnyat madat karnyasathi keela jaato. yaa anuprayogamadhye, sthanik resheey aalekhanche trikon, khand aakar teensah greachy aakrutinche khand tayar karatat. jaalishi sambandhit greek aakruti hypergraphchya paach kinwa tyahoon adhikchya sthanik resheey aalekhatun yetat, udaharanarth dhruviyata aalekhatun tayar kelyapramane.
yadruchchik namune aanee aalekh kadhoon taknyachya lemmache sanyojan aniyantrit 3-samaan rekheey hypergraph kinwa aanshik steiner tripple pranalinmadhye mothe uchch-gher 3-samaan hypergraph shodhanyasathi vaparale jaau shakate. tyanantar hii paddhat 3-samaan rekheey hypergraph aanee aanshik steiner tiheri pranalinchya swatantra sankhyevar lakshanrahitpane ghatt khalchya seema siddha karnyasathi vaparali jaau shakate.

