▁field ) ▁is ▁the ▁Hamiltonian ▁cycle ▁polynomial ▁of ▁its ▁weight ed ▁adj ac ency ▁matrix ▁defined ▁as ▁the ▁sum ▁of ▁the ▁products ▁of ▁the ▁arc ▁weights ▁of ▁the ▁dig raph ' s ▁Hamiltonian ▁cycles . ▁This ▁polynomial ▁is ▁not ▁ident ically ▁zero ▁as ▁a ▁function ▁in ▁the ▁arc ▁weights ▁if ▁and ▁only ▁if ▁the ▁dig raph ▁is ▁Hamiltonian . ▁The ▁relationship ▁between ▁the ▁computational ▁complex ities ▁of ▁computing ▁it ▁and ▁computing ▁the ▁permanent ▁was ▁shown ▁in ▁. ▁ ▁See ▁also ▁▁ ▁Bar net te ' s ▁conject ure , ▁an ▁open ▁problem ▁on ▁Hamilton icity ▁of ▁cub ic ▁b ipart ite ▁poly hed ral ▁graphs ▁ ▁E uler ian ▁path , ▁a ▁path ▁through ▁all ▁edges ▁in ▁a ▁graph ▁ ▁Fle isch ner ' s ▁theorem , ▁on ▁Hamiltonian ▁squares ▁of ▁graphs ▁ ▁Gray ▁code ▁ ▁Gr in berg ' s ▁theorem ▁giving ▁a ▁necessary ▁condition ▁for ▁plan ar ▁graphs ▁to ▁have ▁a ▁Hamiltonian ▁cycle ▁ ▁Hamiltonian ▁path ▁problem , ▁the ▁computational ▁problem ▁of ▁finding ▁Hamiltonian ▁paths ▁ ▁Hy po h amilton ian ▁graph , ▁a ▁non - H amilton ian ▁graph ▁in ▁which ▁every ▁vertex - de leted ▁sub graph ▁is ▁Hamiltonian ▁ ▁Knight ' s ▁tour , ▁a ▁Hamiltonian ▁cycle ▁in ▁the ▁kn ight ' s ▁graph ▁ ▁L CF ▁notation ▁for ▁Hamiltonian ▁cub ic ▁graphs . ▁ ▁Lov ász ▁conject ure ▁that ▁vertex - trans itive ▁graphs ▁are ▁Hamiltonian ▁ ▁Pan cyc lic ▁graph , ▁graphs ▁with ▁cycles ▁of ▁all ▁lengths ▁including ▁a ▁Hamiltonian ▁cycle ▁ ▁Seven ▁Br id ges