▁almost ▁all ▁random ▁graphs ▁are ▁of ▁class ▁ 1 . ▁However , ▁it ▁is ▁N P - complete ▁to ▁determine ▁whether ▁an ▁arbitrary ▁graph ▁is ▁of ▁class ▁ 1 . ▁▁ ▁proved ▁that ▁plan ar ▁graphs ▁of ▁maximum ▁degree ▁at ▁least ▁eight ▁are ▁of ▁class ▁one ▁and ▁conject ured ▁that ▁the ▁same ▁is ▁true ▁for ▁plan ar ▁graphs ▁of ▁maximum ▁degree ▁seven ▁or ▁six . ▁On ▁the ▁other ▁hand , ▁there ▁exist ▁plan ar ▁graphs ▁of ▁maximum ▁degree ▁r anging ▁from ▁two ▁through ▁five ▁that ▁are ▁of ▁class ▁two . ▁The ▁conject ure ▁has ▁since ▁been ▁proven ▁for ▁graphs ▁of ▁maximum ▁degree ▁seven . ▁Br id g eless ▁plan ar ▁cub ic ▁graphs ▁are ▁all ▁of ▁class ▁ 1 ; ▁this ▁is ▁an ▁equivalent ▁form ▁of ▁the ▁four ▁color ▁theorem . ▁ ▁Reg ular ▁graphs ▁A ▁ 1 - factor ization ▁of ▁a ▁k - regular ▁graph , ▁a ▁partition ▁of ▁the ▁edges ▁of ▁the ▁graph ▁into ▁perfect ▁match ings , ▁is ▁the ▁same ▁thing ▁as ▁a ▁k - edge - color ing ▁of ▁the ▁graph . ▁That ▁is , ▁a ▁regular ▁graph ▁has ▁a ▁ 1 - factor ization ▁if ▁and ▁only ▁if ▁it ▁is ▁of ▁class ▁ 1 . ▁ ▁As ▁a ▁special ▁case ▁of ▁this , ▁a ▁ 3 - edge - color ing ▁of ▁a ▁cub ic ▁( 3 - regular ) ▁graph ▁is ▁sometimes ▁called ▁a ▁T ait ▁color ing . ▁ ▁Not ▁every ▁regular ▁graph ▁has ▁a ▁ 1 - factor ization ; ▁for ▁instance , ▁the