▁once . ▁A ▁Hamiltonian ▁cycle ▁( or ▁Hamiltonian ▁circuit ) ▁is ▁a ▁Hamiltonian ▁path ▁that ▁is ▁a ▁cycle . ▁Det erm ining ▁whether ▁such ▁paths ▁and ▁cycles ▁exist ▁in ▁graphs ▁is ▁the ▁Hamiltonian ▁path ▁problem , ▁which ▁is ▁N P - complete . ▁ ▁Hamiltonian ▁paths ▁and ▁cycles ▁are ▁named ▁after ▁William ▁Row an ▁Hamilton ▁who ▁invent ed ▁the ▁ icos ian ▁game , ▁now ▁also ▁known ▁as ▁Hamilton ' s ▁puzz le , ▁which ▁involves ▁finding ▁a ▁Hamiltonian ▁cycle ▁in ▁the ▁edge ▁graph ▁of ▁the ▁d ode c ah ed ron . ▁ ▁Hamilton ▁solved ▁this ▁problem ▁using ▁the ▁ icos ian ▁calculus , ▁an ▁algebraic ▁structure ▁based ▁on ▁roots ▁of ▁unity ▁with ▁many ▁similar ities ▁to ▁the ▁qu atern ions ▁( also ▁invent ed ▁by ▁Hamilton ). ▁ ▁This ▁solution ▁does ▁not ▁general ize ▁to ▁arbitrary ▁graphs . ▁ ▁Despite ▁being ▁named ▁after ▁Hamilton , ▁Hamiltonian ▁cycles ▁in ▁poly hed ra ▁had ▁also ▁been ▁studied ▁a ▁year ▁earlier ▁by ▁Thomas ▁Kirk man , ▁who , ▁in ▁particular , ▁gave ▁an ▁example ▁of ▁a ▁poly hed ron ▁without ▁Hamiltonian ▁cycles . ▁Even ▁earlier , ▁Hamiltonian ▁cycles ▁and ▁paths ▁in ▁the ▁kn ight ' s ▁graph ▁of ▁the ▁ch ess board , ▁the ▁kn ight ' s ▁tour , ▁had ▁been ▁studied ▁in ▁the ▁ 9 th ▁century ▁in ▁Indian ▁mathematics ▁by ▁Rud r ata , ▁and ▁around ▁the ▁same ▁time ▁in ▁Islam ic ▁mathematics ▁by ▁al - Ad li ▁ar - R umi . ▁In ▁ 1 8 th ▁century ▁Europe ,