▁the ▁bound ▁given ▁by ▁V izing ' s ▁theorem . ▁ ▁Other ▁types ▁of ▁edge ▁color ing ▁ ▁The ▁Th ue ▁number ▁of ▁a ▁graph ▁is ▁the ▁number ▁of ▁colors ▁required ▁in ▁an ▁edge ▁color ing ▁meeting ▁the ▁stronger ▁requirement ▁that , ▁in ▁every ▁even - length ▁path , ▁the ▁first ▁and ▁second ▁hal ves ▁of ▁the ▁path ▁form ▁different ▁sequences ▁of ▁colors . ▁ ▁The ▁ar bor icity ▁of ▁a ▁graph ▁is ▁the ▁minimum ▁number ▁of ▁colors ▁required ▁so ▁that ▁the ▁edges ▁of ▁each ▁color ▁have ▁no ▁cycles ▁( r ather ▁than , ▁in ▁the ▁standard ▁edge ▁color ing ▁problem , ▁having ▁no ▁adjacent ▁pairs ▁of ▁edges ). ▁That ▁is , ▁it ▁is ▁the ▁minimum ▁number ▁of ▁for ests ▁into ▁which ▁the ▁edges ▁of ▁the ▁graph ▁may ▁be ▁partition ed ▁into . ▁Un like ▁the ▁chrom atic ▁index , ▁the ▁ar bor icity ▁of ▁a ▁graph ▁may ▁be ▁computed ▁in ▁polynomial ▁time . ▁ ▁List ▁edge - color ing ▁is ▁a ▁problem ▁in ▁which ▁one ▁is ▁given ▁a ▁graph ▁in ▁which ▁each ▁edge ▁is ▁associated ▁with ▁a ▁list ▁of ▁colors , ▁and ▁must ▁find ▁a ▁proper ▁edge ▁color ing ▁in ▁which ▁the ▁color ▁of ▁each ▁edge ▁is ▁drawn ▁from ▁that ▁edge ' s ▁list . ▁The ▁list ▁chrom atic ▁index ▁of ▁a ▁graph ▁ ▁is ▁the ▁smallest ▁number ▁ ▁with ▁the ▁property ▁that , ▁no ▁matter ▁how ▁one ▁cho oses ▁lists ▁of ▁colors ▁for ▁the ▁edges , ▁as ▁long ▁as ▁each ▁edge ▁has ▁at ▁least ▁ ▁colors ▁in ▁its ▁list , ▁then