▁colors ▁are ▁needed . ▁ ▁De fin itions ▁ ▁As ▁with ▁its ▁vertex ▁counter part , ▁an ▁edge ▁color ing ▁of ▁a ▁graph , ▁when ▁mentioned ▁without ▁any ▁qual ification , ▁is ▁always ▁assumed ▁to ▁be ▁a ▁proper ▁color ing ▁of ▁the ▁edges , ▁meaning ▁no ▁two ▁adjacent ▁edges ▁are ▁assigned ▁the ▁same ▁color . ▁ ▁Here , ▁two ▁distinct ▁edges ▁are ▁considered ▁to ▁be ▁adjacent ▁when ▁they ▁share ▁a ▁common ▁vertex . ▁An ▁edge ▁color ing ▁of ▁a ▁graph ▁ ▁may ▁also ▁be ▁thought ▁of ▁as ▁equivalent ▁to ▁a ▁vertex ▁color ing ▁of ▁the ▁line ▁graph ▁, ▁the ▁graph ▁that ▁has ▁a ▁vertex ▁for ▁every ▁edge ▁of ▁ ▁and ▁an ▁edge ▁for ▁every ▁pair ▁of ▁adjacent ▁edges ▁in ▁. ▁ ▁A ▁proper ▁edge ▁color ing ▁with ▁ ▁different ▁colors ▁is ▁called ▁a ▁( pro per ) ▁- edge - color ing . ▁ ▁A ▁graph ▁that ▁can ▁be ▁assigned ▁a ▁- edge - color ing ▁is ▁said ▁to ▁be ▁- edge - color able . ▁The ▁smallest ▁number ▁of ▁colors ▁needed ▁in ▁a ▁( pro per ) ▁edge ▁color ing ▁of ▁a ▁graph ▁ ▁is ▁the ▁chrom atic ▁index , ▁or ▁edge ▁chrom atic ▁number , ▁. ▁The ▁chrom atic ▁index ▁is ▁also ▁sometimes ▁written ▁using ▁the ▁notation ▁; ▁in ▁this ▁notation , ▁the ▁sub script ▁one ▁indicates ▁that ▁edges ▁are ▁one - dimensional ▁objects . ▁ ▁A ▁graph ▁is ▁- edge - chrom atic ▁if ▁its ▁chrom atic ▁index ▁is ▁exactly ▁. ▁The ▁chrom atic ▁index ▁should ▁not ▁be ▁confused ▁with