3 4 ) ▁every ▁plat onic ▁solid , ▁considered ▁as ▁a ▁graph , ▁is ▁Hamiltonian ▁the ▁C ay ley ▁graph ▁of ▁a ▁finite ▁C ox eter ▁group ▁is ▁Hamiltonian ▁ ▁( For ▁more ▁information ▁on ▁Hamiltonian ▁paths ▁in ▁C ay ley ▁graphs , ▁see ▁the ▁Lov ász ▁conject ure ) ▁ ▁Properties ▁ ▁Any ▁Hamiltonian ▁cycle ▁can ▁be ▁converted ▁to ▁a ▁Hamiltonian ▁path ▁by ▁removing ▁one ▁of ▁its ▁edges , ▁but ▁a ▁Hamiltonian ▁path ▁can ▁be ▁extended ▁to ▁Hamiltonian ▁cycle ▁only ▁if ▁its ▁end points ▁are ▁adjacent . ▁ ▁All ▁Hamiltonian ▁graphs ▁are ▁b icon ne cted , ▁but ▁a ▁b icon ne cted ▁graph ▁need ▁not ▁be ▁Hamiltonian ▁( see , ▁for ▁example , ▁the ▁Peters en ▁graph ). ▁ ▁An ▁E uler ian ▁graph ▁G ▁( a ▁connected ▁graph ▁in ▁which ▁every ▁vertex ▁has ▁even ▁degree ) ▁necessarily ▁has ▁an ▁E uler ▁tour , ▁a ▁closed ▁walk ▁passing ▁through ▁each ▁edge ▁of ▁G ▁exactly ▁once . ▁This ▁tour ▁corresponds ▁to ▁a ▁Hamiltonian ▁cycle ▁in ▁the ▁line ▁graph ▁L ( G ), ▁so ▁the ▁line ▁graph ▁of ▁every ▁E uler ian ▁graph ▁is ▁Hamiltonian . ▁Line ▁graphs ▁may ▁have ▁other ▁Hamiltonian ▁cycles ▁that ▁do ▁not ▁correspond ▁to ▁E uler ▁t ours , ▁and ▁in ▁particular ▁the ▁line ▁graph ▁L ( G ) ▁of ▁every ▁Hamiltonian ▁graph ▁G ▁is ▁itself ▁Hamiltonian , ▁regardless ▁of ▁whether ▁the ▁graph ▁G ▁is ▁E uler ian . ▁ ▁A ▁tournament ▁( with ▁more ▁than ▁two ▁vertices ) ▁is ▁Hamiltonian ▁if ▁and ▁only ▁if ▁it ▁is ▁strongly ▁connected