One unproven generalization of this theorem is equivalent to the statement that chi of G is equal to or greater than the Hadwiger number of G.
One unproven generalization of this theorem is equivalent to the statement that chi of G is equal to or greater than the Hadwiger number of G. It is equivalent to the statement that no snark is planar. Another generalization of this theorem was proven by Ringel and Youngs and depends on the floor of an expression involving the square root of 1 plus 48g. It was proven by finding an unavoidable set of reducible configurations, using the method of discharging; a later proof improved on the quartic time algorithm used. It is a special case of the Heawood conjecture. For 10 points, name this theorem proved by Appel and Hankin via computer, which gives the minimum number of shades necessary to produce a map.
One unproven generalization of this theorem is equivalent to the statement that chi of G is equal to or greater than the Hadwiger number of G.
 four-color theorem [or Guthrie's problem]