Simplification and verification Since the proving of the theorem, a new approach has led to both a shorter proof and a more efficient algorithm for 4-coloring maps. In 1996, Neil Robertson, Daniel P. Sanders, Paul Seymour, and Robin Thomas created a quadratic-time algorithm (requiring only O(n2) time, where n is the number of vertices), improving on a quartic-time algorithm based on Appel and Haken's proof.; . The new proof, based on the same ideas, is similar to Appel and Haken's but more efficient because it reduces the complexity of the problem and requires checking only 633 reducible configurations. Both the unavoidability and reducibility parts of this new proof must be executed by a computer and are impractical to check by hand. In 2001, the same authors announced an alternative proof, by proving the snark conjecture.; . This proof remains unpublished, however.
Simplification and verification Since the proving of the theorem, a new approach has led to both a shorter proof and a more efficient algorithm for 4-coloring maps. In 1996, Neil Robertson, Daniel P. Sanders, Paul Seymour, and Robin Thomas created a quadratic-time algorithm (requiring only O(n2) time, where n is the number of vertices), improving on a quartic-time algorithm based on Appel and Haken's proof.; . The new proof, based on the same ,ideas, is similar to Appel and Haken's but more efficient because it reduces the complexity of the problem and requires checking only 633 reducible configurations. Both the unavoidability and reducibility parts of this new proof must be executed by a computer and are impractical to check by hand. In 2001, the same authors announced an alternative proof, by proving the snark conjecture.; . This proof remains unpublished however.