▁the ▁chrom atic ▁number ▁ ▁or ▁, ▁the ▁minimum ▁number ▁of ▁colors ▁needed ▁in ▁a ▁proper ▁vertex ▁color ing ▁of   . ▁ ▁Unless ▁stated ▁otherwise ▁all ▁graphs ▁are ▁assumed ▁to ▁be ▁simple , ▁in ▁contrast ▁to ▁mult ig raph s ▁in ▁which ▁two ▁or ▁more ▁edges ▁may ▁be ▁connecting ▁the ▁same ▁pair ▁of ▁end points ▁and ▁in ▁which ▁there ▁may ▁be ▁self - lo ops . ▁For ▁many ▁problems ▁in ▁edge ▁color ing , ▁simple ▁graphs ▁behave ▁differently ▁from ▁mult ig raph s , ▁and ▁additional ▁care ▁is ▁needed ▁to ▁extend ▁the or ems ▁about ▁edge ▁color ings ▁of ▁simple ▁graphs ▁to ▁the ▁mult ig raph ▁case . ▁ ▁Rel ation ▁to ▁matching ▁ ▁A ▁matching ▁in ▁a ▁graph ▁ ▁is ▁a ▁set ▁of ▁edges , ▁no ▁two ▁of ▁which ▁are ▁adjacent ; ▁a ▁perfect ▁matching ▁is ▁a ▁matching ▁that ▁includes ▁edges ▁touch ing ▁all ▁of ▁the ▁vertices ▁of ▁the ▁graph , ▁and ▁a ▁maximum ▁matching ▁is ▁a ▁matching ▁that ▁includes ▁as ▁many ▁edges ▁as ▁possible . ▁In ▁an ▁edge ▁color ing , ▁the ▁set ▁of ▁edges ▁with ▁any ▁one ▁color ▁must ▁all ▁be ▁non - adj acent ▁to ▁each ▁other , ▁so ▁they ▁form ▁a ▁matching . ▁That ▁is , ▁a ▁proper ▁edge ▁color ing ▁is ▁the ▁same ▁thing ▁as ▁a ▁partition ▁of ▁the ▁graph ▁into ▁dis joint ▁match ings . ▁ ▁If ▁the ▁size ▁of ▁a ▁maximum ▁matching ▁in ▁a ▁given ▁graph ▁is ▁small , ▁then ▁many ▁match ings ▁will ▁be ▁needed ▁in ▁order ▁to ▁cover ▁all ▁of ▁the ▁edges