▁same ▁frequency . ▁When ▁the ▁communic ations ▁network ▁is ▁arranged ▁as ▁a ▁star ▁network , ▁with ▁a ▁single ▁central ▁switch ▁connected ▁by ▁separate ▁fib ers ▁to ▁each ▁of ▁the ▁nodes , ▁the ▁path ▁color ing ▁problem ▁may ▁be ▁mode led ▁exactly ▁as ▁a ▁problem ▁of ▁edge ▁color ing ▁a ▁graph ▁or ▁mult ig raph , ▁in ▁which ▁the ▁communic ating ▁nodes ▁form ▁the ▁graph ▁vertices , ▁pairs ▁of ▁nodes ▁that ▁wish ▁to ▁communicate ▁form ▁the ▁graph ▁edges , ▁and ▁the ▁frequencies ▁that ▁may ▁be ▁used ▁for ▁each ▁pair ▁form ▁the ▁colors ▁of ▁the ▁edge ▁color ing ▁problem . ▁For ▁communic ations ▁networks ▁with ▁a ▁more ▁general ▁tree ▁topology , ▁local ▁path ▁color ing ▁solutions ▁for ▁the ▁star ▁networks ▁defined ▁by ▁each ▁switch ▁in ▁the ▁network ▁may ▁be ▁patch ed ▁together ▁to ▁form ▁a ▁single ▁global ▁solution . ▁ ▁Open ▁problems ▁ ▁list ▁ 2 3 ▁open ▁problems ▁concerning ▁edge ▁color ing . ▁They ▁include : ▁The ▁conject ure ▁of ▁ ▁that ▁the ▁chrom atic ▁index ▁and ▁fraction al ▁index ▁are ▁within ▁one ▁of ▁each ▁other , ▁which ▁would ▁allow ▁the ▁chrom atic ▁index ▁to ▁be ▁approxim ated ▁within ▁one ▁color ▁in ▁polynomial ▁time . ▁Several ▁conject ures ▁of ▁Jakob sen ▁and ▁others ▁on ▁the ▁structure ▁of ▁critical ▁graphs ▁for ▁edge ▁color ing , ▁graphs ▁of ▁class ▁ 2 ▁such ▁that ▁any ▁sub graph ▁either ▁has ▁smaller ▁maximum ▁degree ▁or ▁is ▁of ▁class ▁ 1 . ▁Jakob sen ▁originally ▁conject ured ▁that ▁all ▁critical ▁graphs ▁have ▁an ▁odd ▁number ▁of ▁vertices , ▁but ▁this