False disproofs Four color theorem

generally, simplest, though invalid, counterexamples attempt create 1 region touches other regions. forces remaining regions colored 3 colors. because 4 color theorem true, possible; however, because person drawing map focused on 1 large region, fail notice remaining regions can in fact colored 3 colors.

this trick can generalized: there many maps if colors of regions selected beforehand, becomes impossible color remaining regions without exceeding 4 colors. casual verifier of counterexample may not think change colors of these regions, counterexample appear though valid.

perhaps 1 effect underlying common misconception fact color restriction not transitive: region has colored differently regions touches directly, not regions touching regions touches. if restriction, planar graphs require arbitrarily large numbers of colors.

other false disproofs violate assumptions of theorem in unexpected ways, such using region consists of multiple disconnected parts, or disallowing regions of same color touching @ point.