▁infinite ▁graphs . ▁For ▁instance , ▁Sh annon ' s ▁and ▁V izing ' s ▁the or ems ▁rel ating ▁the ▁degree ▁of ▁a ▁graph ▁to ▁its ▁chrom atic ▁index ▁both ▁general ize ▁straightforward ly ▁to ▁infinite ▁graphs . ▁▁ ▁cons iders ▁the ▁problem ▁of ▁finding ▁a ▁graph ▁drawing ▁of ▁a ▁given ▁cub ic ▁graph ▁with ▁the ▁properties ▁that ▁all ▁of ▁the ▁edges ▁in ▁the ▁drawing ▁have ▁one ▁of ▁three ▁different ▁s lop es ▁and ▁that ▁no ▁two ▁edges ▁lie ▁on ▁the ▁same ▁line ▁as ▁each ▁other . ▁If ▁such ▁a ▁drawing ▁exists , ▁then ▁clearly ▁the ▁s lop es ▁of ▁the ▁edges ▁may ▁be ▁used ▁as ▁colors ▁in ▁a ▁ 3 - edge - color ing ▁of ▁the ▁graph . ▁For ▁instance , ▁the ▁drawing ▁of ▁the ▁utility ▁graph ▁ ▁as ▁the ▁edges ▁and ▁long ▁diag on als ▁of ▁a ▁regular ▁hex agon ▁represents ▁a ▁ 3 - edge - color ing ▁of ▁the ▁graph ▁in ▁this ▁way . ▁As ▁Rich ter ▁shows , ▁a ▁ 3 - regular ▁simple ▁b ipart ite ▁graph , ▁with ▁a ▁given ▁T ait ▁color ing , ▁has ▁a ▁drawing ▁of ▁this ▁type ▁that ▁represents ▁the ▁given ▁color ing ▁if ▁and ▁only ▁if ▁the ▁graph ▁is ▁ 3 - edge - connected . ▁For ▁a ▁non - b ipart ite ▁graph , ▁the ▁condition ▁is ▁a ▁little ▁more ▁complicated : ▁a ▁given ▁color ing ▁can ▁be ▁represented ▁by ▁a ▁drawing ▁if ▁the ▁b ipart ite ▁double ▁cover ▁of ▁the ▁graph ▁is ▁ 3 - edge -