▁V izing ' s ▁theorem ; ▁see ▁Mis ra ▁& ▁G ries ▁edge ▁color ing ▁algorithm . ▁ ▁For ▁mult ig raph s , ▁ ▁present ▁the ▁following ▁algorithm , ▁which ▁they ▁attribute ▁to ▁Eli ▁Up fal . ▁Make ▁the ▁input ▁mult ig raph ▁ ▁E uler ian ▁by ▁adding ▁a ▁new ▁vertex ▁connected ▁by ▁an ▁edge ▁to ▁every ▁odd - deg ree ▁vertex , ▁find ▁an ▁E uler ▁tour , ▁and ▁choose ▁an ▁orientation ▁for ▁the ▁tour . ▁Form ▁a ▁b ipart ite ▁graph ▁ ▁in ▁which ▁there ▁are ▁two ▁copies ▁of ▁each ▁vertex ▁of ▁, ▁one ▁on ▁each ▁side ▁of ▁the ▁b ipart ition , ▁with ▁an ▁edge ▁from ▁a ▁vertex ▁ ▁on ▁the ▁left ▁side ▁of ▁the ▁b ipart ition ▁to ▁a ▁vertex ▁ ▁on ▁the ▁right ▁side ▁of ▁the ▁b ipart ition ▁whenever ▁the ▁orient ed ▁tour ▁has ▁an ▁edge ▁from ▁ ▁to ▁ ▁in ▁. ▁App ly ▁a ▁b ipart ite ▁graph ▁edge ▁color ing ▁algorithm ▁to ▁. ▁Each ▁color ▁class ▁in ▁ ▁corresponds ▁to ▁a ▁set ▁of ▁edges ▁in ▁ ▁that ▁form ▁a ▁sub graph ▁with ▁maximum ▁degree ▁two ; ▁that ▁is , ▁a ▁dis joint ▁union ▁of ▁paths ▁and ▁cycles , ▁so ▁for ▁each ▁color ▁class ▁in ▁ ▁it ▁is ▁possible ▁to ▁form ▁three ▁color ▁classes ▁in ▁. ▁The ▁time ▁for ▁the ▁algorithm ▁is ▁bounded ▁by ▁the ▁time ▁to ▁edge ▁color ▁a ▁b ipart ite ▁graph , ▁ ▁using ▁the ▁algorithm ▁of ▁. ▁The ▁number ▁of ▁colors ▁this ▁algorithm ▁uses ▁is ▁at ▁most ▁, ▁close ▁to