d ▁because ▁each ▁und irect ed ▁edge ▁corresponds ▁to ▁two ▁directed ▁ar cs ▁and ▁thus ▁the ▁degree ▁of ▁a ▁vertex ▁in ▁the ▁directed ▁graph ▁is ▁twice ▁the ▁degree ▁in ▁the ▁und irect ed ▁graph . ▁ ▁Rah man - K ay k ob ad ▁( 2 0 0 5 ) ▁A ▁simple ▁graph ▁with ▁n ▁vertices ▁has ▁a ▁Hamiltonian ▁path ▁if , ▁for ▁every ▁non - adj acent ▁vertex ▁pairs ▁the ▁sum ▁of ▁their ▁degrees ▁and ▁their ▁short est ▁path ▁length ▁is ▁greater ▁than ▁n . ▁ ▁The ▁above ▁theorem ▁can ▁only ▁recognize ▁the ▁existence ▁of ▁a ▁Hamiltonian ▁path ▁in ▁a ▁graph ▁and ▁not ▁a ▁Hamiltonian ▁Cy cle . ▁ ▁Many ▁of ▁these ▁results ▁have ▁analog ues ▁for ▁bal anced ▁b ipart ite ▁graphs , ▁in ▁which ▁the ▁vertex ▁degrees ▁are ▁compared ▁to ▁the ▁number ▁of ▁vertices ▁on ▁a ▁single ▁side ▁of ▁the ▁b ipart ition ▁rather ▁than ▁the ▁number ▁of ▁vertices ▁in ▁the ▁whole ▁graph . ▁ ▁Ex istence ▁of ▁Hamiltonian ▁cycles ▁in ▁plan ar ▁graphs ▁▁▁ ▁Theorem ▁( Wh it ney , ▁ 1 9 3 1 ) ▁A ▁ 4 - connected ▁plan ar ▁tri ang ulation ▁has ▁a ▁Hamiltonian ▁cycle . ▁▁ ▁Theorem ▁( T ut te , ▁ 1 9 5 6 ) ▁A ▁ 4 - connected ▁plan ar ▁graph ▁has ▁a ▁Hamiltonian ▁cycle . ▁ ▁The ▁Hamiltonian ▁cycle ▁polynomial ▁▁ ▁An ▁algebraic ▁representation ▁of ▁the ▁Hamiltonian ▁cycles ▁of ▁a ▁given ▁weight ed ▁dig raph ▁( wh ose ▁ar cs ▁are ▁assigned ▁weights ▁from ▁a ▁certain ▁ground