. ▁ ▁The ▁number ▁of ▁different ▁Hamiltonian ▁cycles ▁in ▁a ▁complete ▁und irect ed ▁graph ▁on ▁n ▁vertices ▁is ▁ ▁and ▁in ▁a ▁complete ▁directed ▁graph ▁on ▁n ▁vertices ▁is ▁. ▁ ▁These ▁counts ▁assume ▁that ▁cycles ▁that ▁are ▁the ▁same ▁apart ▁from ▁their ▁starting ▁point ▁are ▁not ▁counted ▁separately . ▁ ▁Bond y – Ch v át al ▁theorem ▁▁ ▁The ▁best ▁vertex ▁degree ▁character ization ▁of ▁Hamiltonian ▁graphs ▁was ▁provided ▁in ▁ 1 9 7 2 ▁by ▁the ▁Bond y – Ch v át al ▁theorem , ▁which ▁general izes ▁earlier ▁results ▁by ▁G . ▁A . ▁Di rac ▁( 1 9 5 2 ) ▁and ▁ Ø yst ein ▁O re . ▁Both ▁Di rac ' s ▁and ▁O re ' s ▁the or ems ▁can ▁also ▁be ▁derived ▁from ▁P ó sa ' s ▁theorem ▁( 1 9 6 2 ). ▁Hamilton icity ▁has ▁been ▁widely ▁studied ▁with ▁relation ▁to ▁various ▁parameters ▁such ▁as ▁graph ▁density , ▁t ough ness , ▁forb idden ▁sub graph s ▁and ▁distance ▁among ▁other ▁parameters . ▁Di rac ▁and ▁O re ' s ▁the or ems ▁basically ▁state ▁that ▁a ▁graph ▁is ▁Hamiltonian ▁if ▁it ▁has ▁enough ▁edges . ▁ ▁The ▁Bond y – Ch v át al ▁theorem ▁oper ates ▁on ▁the ▁closure ▁cl ( G ) ▁of ▁a ▁graph ▁G ▁with ▁n ▁vertices , ▁obtained ▁by ▁repeatedly ▁adding ▁a ▁new ▁edge ▁u v ▁connecting ▁a ▁non adj acent ▁pair ▁of ▁vertices ▁u ▁and ▁v ▁with ▁ ▁until ▁no ▁more ▁pairs ▁with