▁have ▁different ▁colors . ▁Str ong ▁edge ▁color ing ▁has ▁applications ▁in ▁channel ▁allocation ▁schemes ▁for ▁wireless ▁networks . ▁ ▁A cyc lic ▁edge ▁color ing ▁is ▁the ▁edge - color ing ▁variant ▁of ▁a cyc lic ▁color ing , ▁an ▁edge ▁color ing ▁for ▁which ▁every ▁two ▁color ▁classes ▁form ▁an ▁a cyc lic ▁sub graph ▁( that ▁is , ▁a ▁forest ). ▁The ▁a cyc lic ▁chrom atic ▁index ▁of ▁a ▁graph ▁, ▁denoted ▁by ▁, ▁is ▁the ▁smallest ▁number ▁of ▁colors ▁needed ▁to ▁have ▁a ▁proper ▁a cyc lic ▁edge ▁color ing ▁of ▁. ▁It ▁has ▁been ▁conject ured ▁that ▁, ▁where ▁ ▁is ▁the ▁maximum ▁degree ▁of ▁. ▁Currently ▁the ▁best ▁known ▁bound ▁is ▁. ▁The ▁problem ▁becomes ▁easier ▁when ▁ ▁has ▁large ▁g irth . ▁More ▁specifically , ▁there ▁is ▁a ▁constant ▁ ▁such ▁that ▁if ▁the ▁g irth ▁of ▁ ▁is ▁at ▁least ▁, ▁then ▁. ▁A ▁similar ▁result ▁is ▁that ▁for ▁all ▁ ▁there ▁exists ▁an ▁ ▁such ▁that ▁if ▁ ▁has ▁g irth ▁at ▁least ▁, ▁then ▁. ▁▁ ▁studied ▁ 3 - edge - color ings ▁of ▁cub ic ▁graphs ▁with ▁the ▁additional ▁property ▁that ▁no ▁two ▁b ich rom atic ▁cycles ▁share ▁more ▁than ▁a ▁single ▁edge ▁with ▁each ▁other . ▁He ▁showed ▁that ▁the ▁existence ▁of ▁such ▁a ▁color ing ▁is ▁equivalent ▁to ▁the ▁existence ▁of ▁a ▁drawing ▁of ▁the ▁graph ▁on ▁a ▁three - dimensional ▁integer ▁grid , ▁with ▁edges ▁parallel ▁to ▁the ▁coordinate ▁axes ▁and ▁each ▁axis - parallel ▁line