▁even . ▁However , ▁the ▁inequality ▁ ▁does ▁not ▁fully ▁explain ▁the ▁chrom atic ▁index ▁of ▁every ▁regular ▁graph , ▁because ▁there ▁are ▁regular ▁graphs ▁that ▁do ▁have ▁perfect ▁match ings ▁but ▁that ▁are ▁not ▁k - edge - color able . ▁For ▁instance , ▁the ▁Peters en ▁graph ▁is ▁regular , ▁with ▁ ▁and ▁with ▁ ▁edges ▁in ▁its ▁perfect ▁match ings , ▁but ▁it ▁does ▁not ▁have ▁a ▁ 3 - edge - color ing . ▁ ▁Rel ation ▁to ▁degree ▁ ▁V izing ' s ▁theorem ▁ ▁The ▁edge ▁chrom atic ▁number ▁of ▁a ▁graph ▁ ▁is ▁very ▁closely ▁related ▁to ▁the ▁maximum ▁degree ▁, ▁the ▁largest ▁number ▁of ▁edges ▁incident ▁to ▁any ▁single ▁vertex ▁of ▁. ▁Clear ly , ▁, ▁for ▁if ▁ ▁different ▁edges ▁all ▁meet ▁at ▁the ▁same ▁vertex ▁, ▁then ▁all ▁of ▁these ▁edges ▁need ▁to ▁be ▁assigned ▁different ▁colors ▁from ▁each ▁other , ▁and ▁that ▁can ▁only ▁be ▁possible ▁if ▁there ▁are ▁at ▁least ▁ ▁colors ▁available ▁to ▁be ▁assigned . ▁V izing ' s ▁theorem ▁( named ▁for ▁V ad im ▁G . ▁V izing ▁who ▁published ▁it ▁in ▁ 1 9 6 4 ) ▁states ▁that ▁this ▁bound ▁is ▁almost ▁tight : ▁for ▁any ▁graph , ▁the ▁edge ▁chrom atic ▁number ▁is ▁either ▁ ▁or ▁. ▁When ▁, ▁G ▁is ▁said ▁to ▁be ▁of ▁class ▁ 1 ; ▁otherwise , ▁it ▁is ▁said ▁to ▁be ▁of ▁class ▁ 2 . ▁ ▁Every ▁b ipart ite ▁graph ▁is ▁of ▁class ▁ 1 , ▁and