▁increasing ▁its ▁size , ▁by ▁mer ging ▁pairs ▁of ▁vertices ▁that ▁belong ▁to ▁the ▁same ▁side ▁of ▁the ▁b ipart ition ▁and ▁then ▁adding ▁a ▁small ▁number ▁of ▁additional ▁vertices ▁and ▁edges . ▁Then , ▁if ▁the ▁degree ▁is ▁odd , ▁Al on ▁finds ▁a ▁single ▁perfect ▁matching ▁in ▁near - linear ▁time , ▁assign s ▁it ▁a ▁color , ▁and ▁removes ▁it ▁from ▁the ▁graph , ▁causing ▁the ▁degree ▁to ▁become ▁even . ▁Finally , ▁Al on ▁applies ▁an ▁observation ▁of ▁, ▁that ▁selecting ▁altern ating ▁subsets ▁of ▁edges ▁in ▁an ▁E uler ▁tour ▁of ▁the ▁graph ▁partitions ▁it ▁into ▁two ▁regular ▁sub graph s , ▁to ▁split ▁the ▁edge ▁color ing ▁problem ▁into ▁two ▁smaller ▁sub problem s , ▁and ▁his ▁algorithm ▁solves ▁the ▁two ▁sub problem s ▁recurs ively . ▁The ▁total ▁time ▁for ▁his ▁algorithm ▁is ▁. ▁ ▁For ▁plan ar ▁graphs ▁with ▁maximum ▁degree ▁, ▁the ▁optimal ▁number ▁of ▁colors ▁is ▁again ▁exactly ▁. ▁With ▁the ▁stronger ▁assumption ▁that ▁, ▁it ▁is ▁possible ▁to ▁find ▁an ▁optimal ▁edge ▁color ing ▁in ▁linear ▁time ▁. ▁ ▁For ▁d - regular ▁graphs ▁which ▁are ▁pseudo - random ▁in ▁the ▁sense ▁that ▁their ▁adj ac ency ▁matrix ▁has ▁second ▁largest ▁eigen value ▁( in ▁absolute ▁value ) ▁at ▁most ▁, ▁d ▁is ▁the ▁optimal ▁number ▁of ▁colors ▁. ▁ ▁Alg orithms ▁that ▁use ▁more ▁than ▁the ▁optimal ▁number ▁of ▁colors ▁ ▁and ▁ ▁describe ▁polynomial ▁time ▁algorithms ▁for ▁color ing ▁any ▁graph ▁with ▁ ▁colors , ▁meeting ▁the ▁bound ▁given ▁by