▁Peters en ▁graph ▁does ▁not . ▁More ▁generally ▁the ▁sn arks ▁are ▁defined ▁as ▁the ▁graphs ▁that , ▁like ▁the ▁Peters en ▁graph , ▁are ▁brid g eless , ▁ 3 - regular , ▁and ▁of ▁class ▁ 2 . ▁ ▁According ▁to ▁the ▁theorem ▁of ▁, ▁every ▁b ipart ite ▁regular ▁graph ▁has ▁a ▁ 1 - factor ization . ▁The ▁theorem ▁was ▁stated ▁earlier ▁in ▁terms ▁of ▁project ive ▁configurations ▁and ▁was ▁proven ▁by ▁Ernst ▁Ste init z . ▁ ▁Mult ig raph s ▁ ▁For ▁mult ig raph s , ▁in ▁which ▁multiple ▁parallel ▁edges ▁may ▁connect ▁the ▁same ▁two ▁vertices , ▁results ▁that ▁are ▁similar ▁to ▁but ▁we aker ▁than ▁V izing ' s ▁theorem ▁are ▁known ▁rel ating ▁the ▁edge ▁chrom atic ▁number ▁ ▁, ▁the ▁maximum ▁degree ▁, ▁and ▁the ▁multip licity ▁, ▁the ▁maximum ▁number ▁of ▁edges ▁in ▁any ▁bundle ▁of ▁parallel ▁edges . ▁As ▁a ▁simple ▁example ▁showing ▁that ▁V izing ' s ▁theorem ▁does ▁not ▁general ize ▁to ▁mult ig raph s , ▁consider ▁a ▁Sh annon ▁mult ig raph , ▁a ▁mult ig raph ▁with ▁three ▁vertices ▁and ▁three ▁bund les ▁of ▁ ▁parallel ▁edges ▁connecting ▁each ▁of ▁the ▁three ▁pairs ▁of ▁vertices . ▁In ▁this ▁example , ▁ ▁( each ▁vertex ▁is ▁incident ▁to ▁only ▁two ▁out ▁of ▁the ▁three ▁bund les ▁of ▁ ▁parallel ▁edges ) ▁but ▁the ▁edge ▁chrom atic ▁number ▁is ▁ ▁( there ▁are ▁ ▁edges ▁in ▁total , ▁and ▁every ▁two ▁edges ▁are ▁adjacent , ▁so ▁all