# Four Color Theorem

## Theorem

Any planar graph $G$ can be assigned a proper vertex $k$-coloring such that $k \le 4$.

## Proof

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## Also known as

In British English, this theorem is spelled **four colour theorem**.

Some sources continue to refer to it as the **four colo(u)r problem**, acknowledging the controversial nature of its supposed proof.

## Also see

## Historical Note

The question of the minimal number of colors that are needed to color any arbitrary planar graph was first considered by Francis Guthrie in $1852$, when he was coloring a map of the counties of England.

He conjectured that the answer was $4$.

He published this conjecture in $1878$.

In $1879$, Alfred Bray Kempe published what he believed to be a proof, but in $1890$ it was shown by Percy John Heawood to be flawed.

In that same year, Heawood proved that $5$ colors were certainly enough.

In $1880$, another proof was published, but that was also shown to be flawed.

During the course of the century following, many developments in graph theory were made as a result of research into this problem.

What is now believe to be a sound proof of the Four Color Theorem was presented in $1976$ by Kenneth Ira Appel and Wolfgang Haken.

Their proof relies heavily on a computer program, set up to search through all the various cases.

Hence their proof does not easily conform to $\mathsf{Pr} \infty \mathsf{fWiki}$'s format.

Apart from this computer-based search, the proof is similar to that of the Five Color Theorem, a related but weaker result.

Some mathematicians distrust this proof because of its reliance on computers, and its consequent inability to be checked by humans.

This is part of a broader debate in mathematics over the increasing use of computers in proofs.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**four-colour problem** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**four-colour problem** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**chromatic number** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Four Colour Theorem**