
menaj prashnam

kompinettoriyal maathamaattiksil, menej prashnam allengil problem dess menejas oru vatta meshayil oru koottam purusha-sthree dambathikale irikkaan kazhiyunna vividha vazhikalude ennam chodikkunnu, angane purushanmaarum sthreekalum marimari irikkukayum aarum avarude pankaaliyude arikil irikkaathirikkukayum cheiyunnu. (oru purusha-sthree dambathikale soochippikkunna "veedu" ennathinte franch padamaanu "menej".) 1891ll edward lukes swathanthramaayi, kurachu varshangalkk munbu, nottu sidhaanthavumaayi bandhappett peettar gutri taitum chernnaanu eee prashnam roopappeduthiyathu. 3,4,5, ennivaykku thulyamaaya niravadhi dambathikalkku... irippida krameekaranangalude ennam
eee samkhyakalum akkangalude anubandha kramangalum kanakkukoottunnathinaayi ganithashaasthranjar soothravaakyangalum aavarthana samavaakyangalum vikasippicheduthittundu. maryada, nottu sidhaantham ennivayilekkulla prayogangalkkoppam, eee samkhyakalkku oru graph saidhaanthika vyaakhyaanavumundu avaa graafukalude chila kudumbangalile poruthangaludeyum hamiltonian chakrangaludeyum ennam kanakkakkunnu.
touchardinte formula.
m" n "ennathu" n "dambathikalkkulla irippida krameekaranangalude ennathe soochippikkatte. soothravaakyam uruthirinju
thudarnnulla niravadhi pravartthanangal eee soothravaakyathinte badal thelivukalilekkum prashnatthinte vividha saamaanyavalkkaricha pathippukalilekkum pooyi. 
aadya tharatthilulla chebishev polynomialukal ulppedunna "m" n "-nuu vyathyasthamaaya oru umbral formula nalki.
menej nambarukalum ladys phastt solyooshanukalum.
2 × "n" undu! sthreekalkku irikkaanulla vazhikal sthreekalkku krameekarikkaan kazhiyunna rund settu seettukal undu, koodaathe "enn" undu! oru prathyeka settu seettukalil avare irikkaanulla vazhikal. sthreekalkkulla oro irippida krameekaranatthinum, undu
purushanmaare irikkaanulla vazhikal; eee soothravaakyam 2x "n" ozhivaakkunnu! touchardinte formula ninnulla ghadakam. thathfalamaayundaakunna cheria samkhyakal (veendum, "n" = 3 ll ninnu aarambikkunnu),
avaye menej nambarukal ennu vilikkunnu. aduthulla seettukalude overlapping jody roopeekarikkunnathinulla vazhikalude ennamo allengil thulyamaayi, konukalude saikkil graphile arikukalude poruthangalude ennamo aanu factor formula _ 3. oru porutthatthinte oro attatthum irikkunna aalukal dambathikalaayirikkenda krameekaranangalil ulppeduthal-ozhivakkal thathwam prayogikkunnathinte pettennulla falamaanu 'phoor' enna prayogam.
eee prashnathinulla pariharangal aadyam sthreekalkkulla ellah irippida krameekaranangalum kandethukayum thudarnnu eee bhagika irippida krameekaranangalkk oronninum, purushanmaare avarude pankaalikalil ninnu akatti iruthi poortthiyaakkaanulla vazhikalude ennam kanakkaakkukayum cheythu. sthreekalude pankaalittham kanakkiledukkunnathinum ellah irippida krameekaranangalum oresamayam pariganikkunnathiloode touchardinte soothravaakyam nerittu uruthirinjekkaamennu bogart doyal vaadichu. ennirunnaalum, chila bogartinteyum doyilinteyum ashayangal upayogichukondu (lingabhedam illaatha bhashayil vaadathe punarnirmikkaan avar shradhichenkilum) mukalil vivaricha kooduthal neraaya sthree-aadya pariharam kandethi.
menej nambarukal aavarthana bandhathe thripthippeduthunnu
lalithamaaya naalu-tame aavarthanavum
athil ninnu menej nambarukal eluppathil kanakkakkaan kazhiyum.
graph-saidhaanthika vyaakhyaanangal.
kireeda graafukalile hamiltonian chakrangal pole, menej prashnathinulla pariharangal graph-saidhaanthika padangalil vyaakhyaanikkaam. kn, n" enna samboornna ubhaykakshi graphil ninnu oru thikanja poruttham neekkam cheythaanu oru kireeda graph roopappedunnathu; ithinu rund nirangalude 2 "n" konukalundu, koodaathe oru niratthinte oro konum matte niratthinte konukalilonnozhike matellaavarumaayum bandhippichirikkunnu. menej prashnatthinte kaaryatthil, graafinte shringangal purushanmareyum sthreekaleyum prathinidheekarikkunnu, arikukal parasparam aduthu irikkaan anuvaadamulla purushanmaarudeyum sthreekaludeyum jodikale prathinidheekarikkunnu. oro purushan oro sthreeyumaayum bandhippikkunna oru samboornna ubhaykakshi graphil ninnu purusha-sthree dambathikal roopeekaricha thikanja poruttham neekkam cheythaanu eee graph roopappedunnathu. meshaykk chuttumulla aalukalude kramam anusarichu saadhuvaaya eathu irippida krameekaranavum vivarikkaam, ithu graphile oru hamiltonian saikkil roopappeduthunnu. ennirunnaalum, rund hamiltonian chakrangal arambha sheershakam pariganikkathe ore chaakrika kramatthil ore konukale bandhippikkukayaanenkil thulyamaayi kanakkaakkappedunnu, athesamayam menej prashnatthil arambha sthaanam praadhaanyamarhikkunnu aalisinte tea paarttiyilennapole, ellah athidhikalum avarude sthaanangal oru seettil maattukayaanengil, ore saikkil vivarichittundenkilum athu vyathyastha irippida krameekaranamaayi kanakkaakkappedunnu. athinaal, oru kireeda graphile ooriyantad hamiltonian saikkilukalude ennam irippida krameekaranangalude ennathekkaal 2 'n' enna ghadakam cheruthaanu, ennaal ('n' −1) enna ghadakam kondu valuthaanu! menaj nambarukalekkaal. eee graafukalile chakrangalude samkhyakalude kramam (munbatheppole, "n" = 3 ll aarambikkunnu)
prashnatthinte randaamathe graph-saidhaanthika vivaranavum saadhyamaan. sthreekal irunnukazhinjaal, bakkiyulla purushanmarkkulla saadhyamaaya irippida krameekaranangale oru samboornna ubhaykakshi graphil ninnu orotta hamiltonian saikkil neekkam cheythukondu roopeekaricha graphil thikanja poruthappedutthalukalaayi vivarikkaam; graphil purushanmarumaayi thuranna seettukale bandhippikkunna arikukalundu, saikkil neekkam cheiyunnathu purushanmaare avarude bhaaryamaarod chernnulla eathenkilum thuranna seettukalil irikkunnathu vilakkunnathinodu yojikkunnu. oru ubhaykakshi graphile porutthangal ennunna prashnam, athinaal menej nambarukal kanakkukoottunna prashnam "oru fortiori", chila 0-1 maatriksukalude permanantukal upayogichu pariharikkan kazhiyum. menej prashnatthinte kaaryatthil, prashnatthinte eee kazhchappadil ninnu undaakunna matricks sarkkulant maatriksaanu, athil uthppaadippikkunna variyude rund aduthulla gadakangal ozhike ellaam 1 nuu thulyamaanu.
nottu sidhaantham.
menej prashnam padikkunnathinulla taittinte prachodanam "enn" ennu parayappedunna oru nischitha ennam crossingukalulla ganithashaasthra kanangalude poornnamaaya pattika kandethaan sramikkunnathil ninnaan. nottu diagramukalkkulla daukkar notteshanil, athinte aadyakaala roopam taitt upayogichirunnu, oru nottu svayam kadakkunna 2 "enn" pooyintukal, kettinoppam thudarchayaaya kramatthil, 1 muthal 2 "enn" vareyulla 2 "enn" samkhyakal upayogichu label cheythirikkunnu. oru kuranja rekhaachithrathil, oru crossingile rund labelukal thudarchayaayi aakaan kazhiyilla, athinaal oro crossingileyum jody labelukalude koottam, nottine prathinideekarikkaan daukkar notteshanil upayogikkunnu, 1 muthal 2 "n" vareyulla shreniyilulla oro samkhyaykkum oru sheershakavum vyathyastha samathvavum thudarchayaaya modulo 2 "n" allatha oro jody samkhyakalkkum edayilulla oru arikumulla oru graphil oru thikanja poruthamaayi vyaakhyaanikkaam. oru samboornna ubhaykakshi graphil ninnu (ellah jody samkhyakaleyum vyathyastha samathvavumaayi bandhippikkunna) oru hamiltonian saikkil (thudarchayaaya samkhyakale bandhippikkunna) neekkam cheythaanu eee graph roopappedunnathu, athinaal ithinu oru menej nambarinu thulyamaaya niravadhi poruthangalundu. alternating notsinu, eee poruthappedutthal nottu diagram thanne vivarikkan mathiyakum; mattu notsinu, crossinginte rund straandukalil eathaanu matte strandinu mukalilullathennu nirnnayikkaan oro crossing jodiykkum oru adhika positteev allengil negatteevu adayaalam vyakthamaakkendathundu.
ennirunnaalum, nottu listing prashnathinu menej prashnatthil illaatha chila adhika samamithikalundu oraal mattoru crossing pooyintil labeling aarambhikkukayaanenkil ore nottu diagraminu vyathyastha daukkar notteshanukal labikkum, eee vyathyastha notteshanukalellaam ore diagramine prathinideekarikkunnathaayi kanakkaakkanam. ekkaaranathaal, oru chaakrika kramappedutthal vazhi parasparam vyathyaasappedunna rund porutthangal thulyamaayi kanakkaakkukayum oru thavana maathram kanakkaakkukayum venam. vyathyastha poruthangalude ennam kaanikkunna eee parishkaricha enumeration prashnam pariharichu

