▁of ▁the ▁graph . ▁Ex pressed ▁more ▁formally , ▁this ▁reasoning ▁implies ▁that ▁if ▁a ▁graph ▁has ▁ ▁edges ▁in ▁total , ▁and ▁if ▁at ▁most ▁ ▁edges ▁may ▁belong ▁to ▁a ▁maximum ▁matching , ▁then ▁every ▁edge ▁color ing ▁of ▁the ▁graph ▁must ▁use ▁at ▁least ▁ ▁different ▁colors . ▁For ▁instance , ▁the ▁ 1 6 - ver tex ▁plan ar ▁graph ▁shown ▁in ▁the ▁illustr ation ▁has ▁ ▁edges . ▁In ▁this ▁graph , ▁there ▁can ▁be ▁no ▁perfect ▁matching ; ▁for , ▁if ▁the ▁center ▁vertex ▁is ▁matched , ▁the ▁remaining ▁un match ed ▁vertices ▁may ▁be ▁grouped ▁into ▁three ▁different ▁connected ▁components ▁with ▁four , ▁five , ▁and ▁five ▁vertices , ▁and ▁the ▁components ▁with ▁an ▁odd ▁number ▁of ▁vertices ▁cannot ▁be ▁perfectly ▁matched . ▁However , ▁the ▁graph ▁has ▁maximum ▁match ings ▁with ▁seven ▁edges , ▁so ▁. ▁Therefore , ▁the ▁number ▁of ▁colors ▁needed ▁to ▁edge - color ▁the ▁graph ▁is ▁at ▁least ▁ 2 4 / 7 , ▁and ▁since ▁the ▁number ▁of ▁colors ▁must ▁be ▁an ▁integer ▁it ▁is ▁at ▁least ▁four . ▁ ▁For ▁a ▁regular ▁graph ▁of ▁degree ▁ ▁that ▁does ▁not ▁have ▁a ▁perfect ▁matching , ▁this ▁lower ▁bound ▁can ▁be ▁used ▁to ▁show ▁that ▁at ▁least ▁ ▁colors ▁are ▁needed . ▁In ▁particular , ▁this ▁is ▁true ▁for ▁a ▁regular ▁graph ▁with ▁an ▁odd ▁number ▁of ▁vertices ▁( such ▁as ▁the ▁odd ▁complete ▁graphs ); ▁for ▁such ▁graphs , ▁by ▁the ▁hand sh aking ▁lemma , ▁ ▁must ▁itself ▁be