▁but ▁not ▁quite ▁the ▁same ▁as ▁Sh annon ' s ▁bound ▁of ▁. ▁It ▁may ▁also ▁be ▁made ▁into ▁a ▁parallel ▁algorithm ▁in ▁a ▁straightforward ▁way . ▁In ▁the ▁same ▁paper , ▁Kar lo ff ▁and ▁Sh mo ys ▁also ▁present ▁a ▁linear ▁time ▁algorithm ▁for ▁color ing ▁mult ig raph s ▁of ▁maximum ▁degree ▁three ▁with ▁four ▁colors ▁( match ing ▁both ▁Sh annon ' s ▁and ▁V izing ' s ▁bounds ) ▁that ▁oper ates ▁on ▁similar ▁principles : ▁their ▁algorithm ▁adds ▁a ▁new ▁vertex ▁to ▁make ▁the ▁graph ▁E uler ian , ▁finds ▁an ▁E uler ▁tour , ▁and ▁then ▁cho oses ▁altern ating ▁sets ▁of ▁edges ▁on ▁the ▁tour ▁to ▁split ▁the ▁graph ▁into ▁two ▁sub graph s ▁of ▁maximum ▁degree ▁two . ▁The ▁paths ▁and ▁even ▁cycles ▁of ▁each ▁sub graph ▁may ▁be ▁colored ▁with ▁two ▁colors ▁per ▁sub graph . ▁After ▁this ▁step , ▁each ▁remaining ▁odd ▁cycle ▁contains ▁at ▁least ▁one ▁edge ▁that ▁may ▁be ▁colored ▁with ▁one ▁of ▁the ▁two ▁colors ▁belonging ▁to ▁the ▁opposite ▁sub graph . ▁Rem oving ▁this ▁edge ▁from ▁the ▁odd ▁cycle ▁leaves ▁a ▁path , ▁which ▁may ▁be ▁colored ▁using ▁the ▁two ▁colors ▁for ▁its ▁sub graph . ▁ ▁A ▁gre edy ▁color ing ▁algorithm ▁that ▁cons iders ▁the ▁edges ▁of ▁a ▁graph ▁or ▁mult ig raph ▁one ▁by ▁one , ▁assigning ▁each ▁edge ▁the ▁first ▁available ▁color , ▁may ▁sometimes ▁use ▁as ▁many ▁as ▁ ▁colors , ▁which ▁may ▁be ▁nearly ▁twice ▁as ▁many ▁number ▁of ▁colors ▁as ▁is