▁necessary . ▁However , ▁it ▁has ▁the ▁advantage ▁that ▁it ▁may ▁be ▁used ▁in ▁the ▁online ▁algorithm ▁setting ▁in ▁which ▁the ▁input ▁graph ▁is ▁not ▁known ▁in ▁advance ; ▁in ▁this ▁setting , ▁its ▁compet itive ▁ratio ▁is ▁two , ▁and ▁this ▁is ▁optimal : ▁no ▁other ▁online ▁algorithm ▁can ▁achieve ▁a ▁better ▁performance . ▁However , ▁if ▁edges ▁arrive ▁in ▁a ▁random ▁order , ▁and ▁the ▁input ▁graph ▁has ▁a ▁degree ▁that ▁is ▁at ▁least ▁log arith mic , ▁then ▁smaller ▁compet itive ▁r ati os ▁can ▁be ▁achieved . ▁ ▁Several ▁authors ▁have ▁made ▁conject ures ▁that ▁imply ▁that ▁the ▁fraction al ▁chrom atic ▁index ▁of ▁any ▁mult ig raph ▁( a ▁number ▁that ▁can ▁be ▁computed ▁in ▁polynomial ▁time ▁using ▁linear ▁programming ) ▁is ▁within ▁one ▁of ▁the ▁chrom atic ▁index . ▁If ▁these ▁conject ures ▁are ▁true , ▁it ▁would ▁be ▁possible ▁to ▁compute ▁a ▁number ▁that ▁is ▁never ▁more ▁than ▁one ▁off ▁from ▁the ▁chrom atic ▁index ▁in ▁the ▁mult ig raph ▁case , ▁matching ▁what ▁is ▁known ▁via ▁V izing ' s ▁theorem ▁for ▁simple ▁graphs . ▁Although ▁un pro ven ▁in ▁general , ▁these ▁conject ures ▁are ▁known ▁to ▁hold ▁when ▁the ▁chrom atic ▁index ▁is ▁at ▁least ▁, ▁as ▁can ▁happen ▁for ▁mult ig raph s ▁with ▁sufficiently ▁large ▁multip licity . ▁ ▁Ex act ▁algorithms ▁It ▁is ▁straightforward ▁to ▁test ▁whether ▁a ▁graph ▁may ▁be ▁edge ▁colored ▁with ▁one ▁or ▁two ▁colors , ▁so ▁the ▁first ▁n ont rivial ▁case ▁of ▁edge ▁color ing