▁this ▁property ▁can ▁be ▁found . ▁ ▁Bond y – Ch v át al ▁theorem ▁( 1 9 7 6 ) ▁A ▁graph ▁is ▁Hamiltonian ▁if ▁and ▁only ▁if ▁its ▁closure ▁is ▁Hamiltonian . ▁ ▁As ▁complete ▁graphs ▁are ▁Hamiltonian , ▁all ▁graphs ▁whose ▁closure ▁is ▁complete ▁are ▁Hamiltonian , ▁which ▁is ▁the ▁content ▁of ▁the ▁following ▁earlier ▁the or ems ▁by ▁Di rac ▁and ▁O re . ▁ ▁Di rac ▁( 1 9 5 2 ) ▁A ▁simple ▁graph ▁with ▁n ▁vertices ▁() ▁is ▁Hamiltonian ▁if ▁every ▁vertex ▁has ▁degree ▁ ▁or ▁greater . ▁ ▁O re ▁( 1 9 6 0 ) ▁A ▁simple ▁graph ▁with ▁n ▁vertices ▁( n ▁ ≥ ▁ 3 ) ▁is ▁Hamiltonian ▁if , ▁for ▁every ▁pair ▁of ▁non - adj acent ▁vertices , ▁the ▁sum ▁of ▁their ▁degrees ▁is ▁n ▁or ▁greater ▁( see ▁O re ' s ▁theorem ). ▁ ▁The ▁following ▁the or ems ▁can ▁be ▁regarded ▁as ▁directed ▁versions : ▁ ▁G hou ila - H ou iri ▁( 1 9 6 0 ) ▁A ▁strongly ▁connected ▁simple ▁directed ▁graph ▁with ▁n ▁vertices ▁is ▁Hamiltonian ▁if ▁every ▁vertex ▁has ▁a ▁full ▁degree ▁greater ▁than ▁or ▁equal ▁to   n . ▁ ▁Me yn iel ▁( 1 9 7 3 ) ▁A ▁strongly ▁connected ▁simple ▁directed ▁graph ▁with ▁n ▁vertices ▁is ▁Hamiltonian ▁if ▁the ▁sum ▁of ▁full ▁degrees ▁of ▁every ▁pair ▁of ▁distinct ▁non - adj acent ▁vertices ▁is ▁greater ▁than ▁or ▁equal ▁to ▁. ▁ ▁The ▁number ▁of ▁vertices ▁must ▁be ▁double