▁a ▁color ing ▁is ▁guaranteed ▁to ▁be ▁possible . ▁Thus , ▁the ▁list ▁chrom atic ▁index ▁is ▁always ▁at ▁least ▁as ▁large ▁as ▁the ▁chrom atic ▁index . ▁The ▁D init z ▁conject ure ▁on ▁the ▁completion ▁of ▁partial ▁Latin ▁squares ▁may ▁be ▁re phr ased ▁as ▁the ▁statement ▁that ▁the ▁list ▁edge ▁chrom atic ▁number ▁of ▁the ▁complete ▁b ipart ite ▁graph ▁ ▁equals ▁its ▁edge ▁chrom atic ▁number ,   . ▁ ▁resolved ▁the ▁conject ure ▁by ▁prov ing , ▁more ▁generally , ▁that ▁in ▁every ▁b ipart ite ▁graph ▁the ▁chrom atic ▁index ▁and ▁list ▁chrom atic ▁index ▁are ▁equal . ▁The ▁equality ▁between ▁the ▁chrom atic ▁index ▁and ▁the ▁list ▁chrom atic ▁index ▁has ▁been ▁conject ured ▁to ▁hold , ▁even ▁more ▁generally , ▁for ▁arbitrary ▁mult ig raph s ▁with ▁no ▁self - lo ops ; ▁this ▁conject ure ▁remains ▁open . ▁ ▁Many ▁other ▁commonly ▁studied ▁variations ▁of ▁vertex ▁color ing ▁have ▁also ▁been ▁extended ▁to ▁edge ▁color ings . ▁For ▁instance , ▁complete ▁edge ▁color ing ▁is ▁the ▁edge - color ing ▁variant ▁of ▁complete ▁color ing , ▁a ▁proper ▁edge ▁color ing ▁in ▁which ▁each ▁pair ▁of ▁colors ▁must ▁be ▁represented ▁by ▁at ▁least ▁one ▁pair ▁of ▁adjacent ▁edges ▁and ▁in ▁which ▁the ▁goal ▁is ▁to ▁maxim ize ▁the ▁total ▁number ▁of ▁colors . ▁Str ong ▁edge ▁color ing ▁is ▁the ▁edge - color ing ▁variant ▁of ▁strong ▁color ing , ▁an ▁edge ▁color ing ▁in ▁which ▁every ▁two ▁edges ▁with ▁adjacent ▁end points ▁must