Dürre-Heesch algorithm used to prove the four-color theorem.
The essence of the proof is that if we assume the existence of a counterexample, then it must contain at least one of 2822 specific configurations, as shown by John P. Steinberger. Furthermore, it has also been proven by an infinite descent argument that a smallest counterexample cannot contain any D- reducible configurations. Thus, if we show that all of the above mentioned configurations are D-reducible, then the four color theorem holds.