▁is ▁testing ▁whether ▁a ▁graph ▁has ▁a ▁ 3 - edge - color ing . ▁As ▁ ▁showed , ▁it ▁is ▁possible ▁to ▁test ▁whether ▁a ▁graph ▁has ▁a ▁ 3 - edge - color ing ▁in ▁time ▁, ▁while ▁using ▁only ▁polynomial ▁space . ▁Although ▁this ▁time ▁bound ▁is ▁exponential , ▁it ▁is ▁significantly ▁faster ▁than ▁a ▁br ute ▁force ▁search ▁over ▁all ▁possible ▁assign ments ▁of ▁colors ▁to ▁edges . ▁Every ▁b icon ne cted ▁ 3 - regular ▁graph ▁with ▁ ▁vertices ▁has ▁▁ 3 - edge - color ings ; ▁all ▁of ▁which ▁can ▁be ▁listed ▁in ▁time ▁ ▁( some what ▁slower ▁than ▁the ▁time ▁to ▁find ▁a ▁single ▁color ing ); ▁as ▁Greg ▁K uper berg ▁observed , ▁the ▁graph ▁of ▁a ▁pr ism ▁over ▁an ▁- s ided ▁polygon ▁has ▁ ▁color ings ▁( lower ▁instead ▁of ▁upper ▁bound ), ▁showing ▁that ▁this ▁bound ▁is ▁tight . ▁ ▁By ▁applying ▁exact ▁algorithms ▁for ▁vertex ▁color ing ▁to ▁the ▁line ▁graph ▁of ▁the ▁input ▁graph , ▁it ▁is ▁possible ▁to ▁optim ally ▁edge - color ▁any ▁graph ▁with ▁ ▁edges , ▁regardless ▁of ▁the ▁number ▁of ▁colors ▁needed , ▁in ▁time ▁ ▁and ▁exponential ▁space , ▁or ▁in ▁time ▁ ▁and ▁only ▁polynomial ▁space ▁. ▁ ▁Because ▁edge ▁color ing ▁is ▁N P - complete ▁even ▁for ▁three ▁colors , ▁it ▁is ▁unlikely ▁to ▁be ▁fixed ▁parameter ▁tract able ▁when ▁paramet riz ed ▁by ▁the ▁number ▁of ▁colors . ▁However , ▁it ▁is ▁tract able ▁for ▁other