▁parameters . ▁In ▁particular , ▁ ▁showed ▁that ▁for ▁graphs ▁of ▁tre ew idth ▁, ▁an ▁optimal ▁edge ▁color ing ▁can ▁be ▁computed ▁in ▁time ▁, ▁a ▁bound ▁that ▁depends ▁sup ere x pon entially ▁on ▁ ▁but ▁only ▁linear ly ▁on ▁the ▁number ▁ ▁of ▁vertices ▁in ▁the ▁graph . ▁▁ ▁form ulate ▁the ▁edge ▁color ing ▁problem ▁as ▁an ▁integer ▁program ▁and ▁describe ▁their ▁experience ▁using ▁an ▁integer ▁programming ▁sol ver ▁to ▁edge ▁color ▁graphs . ▁However , ▁they ▁did ▁not ▁perform ▁any ▁complexity ▁analysis ▁of ▁their ▁algorithm . ▁ ▁Add itional ▁properties ▁ ▁A ▁graph ▁is ▁uniqu ely ▁- edge - color able ▁if ▁there ▁is ▁only ▁one ▁way ▁of ▁partition ing ▁the ▁edges ▁into ▁ ▁color ▁classes , ▁ign oring ▁the ▁ ▁possible ▁permut ations ▁of ▁the ▁colors . ▁For ▁, ▁the ▁only ▁uniqu ely ▁- edge - color able ▁graphs ▁are ▁paths , ▁cycles , ▁and ▁stars , ▁but ▁for ▁ ▁other ▁graphs ▁may ▁also ▁be ▁uniqu ely ▁- edge - color able . ▁Every ▁uniqu ely ▁ 3 - edge - color able ▁graph ▁has ▁exactly ▁three ▁Hamiltonian ▁cycles ▁( formed ▁by ▁deleting ▁one ▁of ▁the ▁three ▁color ▁classes ) ▁but ▁there ▁exist ▁ 3 - regular ▁graphs ▁that ▁have ▁three ▁Hamiltonian ▁cycles ▁and ▁are ▁not ▁uniqu ely ▁ 3 - color able , ▁such ▁as ▁the ▁generalized ▁Peters en ▁graphs ▁ ▁for ▁. ▁The ▁only ▁known ▁non plan ar ▁uniqu ely ▁ 3 - color able ▁graph ▁is ▁the ▁generalized ▁Peters en ▁graph ▁, ▁and ▁it