▁was ▁eventually ▁dis pro ved . ▁Several ▁other ▁conject ures ▁weak ening ▁this ▁one , ▁or ▁bound ing ▁the ▁numbers ▁of ▁vertices ▁of ▁critical ▁graphs ▁and ▁critical ▁mult ig raph s , ▁remain ▁open . ▁V izing ' s ▁problem ▁of ▁class ifying ▁the ▁maximum ▁degrees ▁that ▁are ▁possible ▁for ▁class ▁ 2 ▁plan ar ▁graphs . ▁The ▁over full ▁sub graph ▁conject ure ▁of ▁A . ▁J . ▁W . ▁Hil ton , ▁stating ▁that ▁graphs ▁with ▁degree ▁at ▁least ▁ ▁are ▁either ▁of ▁class ▁ 1 ▁or ▁contain ▁a ▁sub graph ▁with ▁the ▁same ▁degree ▁ ▁as ▁the ▁original ▁graph , ▁and ▁with ▁an ▁odd ▁number ▁ ▁of ▁vertices , ▁such ▁that ▁the ▁number ▁of ▁edges ▁in ▁the ▁sub graph ▁is ▁greater ▁than ▁, ▁and ▁a ▁similar ▁conject ure ▁by ▁Herbert ▁Gr öt z sch ▁and ▁Paul ▁Se ym our ▁concerning ▁plan ar ▁graphs ▁in ▁place ▁of ▁high - deg ree ▁graphs . ▁A ▁conject ure ▁of ▁Am anda ▁Ch et wy nd ▁and ▁Anthony ▁Hil ton ▁( poss ibly ▁going ▁back ▁earlier ▁to ▁the ▁work ▁of ▁Gabriel ▁Andrew ▁Di rac ) ▁that ▁regular ▁graphs ▁with ▁an ▁even ▁number ▁ ▁of ▁vertices ▁and ▁with ▁degree ▁at ▁least ▁ ▁are ▁of ▁class ▁ 1 . ▁A ▁conject ure ▁of ▁Claude ▁Ber ge ▁and ▁D . ▁R . ▁Ful k erson ▁that ▁the ▁ 6 - regular ▁mult ig raph s ▁formed ▁by ▁dou bling ▁every ▁edge ▁of ▁a ▁brid g eless ▁ 3 - regular ▁simple ▁graph ▁may ▁be ▁edge - color