The Four Color Theorem and Maps?

Four color theorem

What Is the Four Color Theorem? 

The Four Color Theorem is a famous result in graph theory, which states that any map on a plane can be colored using only four colors in such a way that no two adjacent regions (countries) share the same color. The theorem does not apply to maps on surfaces other than a plane.

This is mostly used by Cartographers, which are involved at map-making of various stages  including liaising with clients, carrying out research and checking accuracy, as well as designing maps themselves. 

The Story and Debates on the Theory

The story of the Four Color Theorem spans several centuries, with contributions from various mathematicians. Here’s a brief overview:

The origins of the problem can be traced back to the 19th century. Mathematicians such as Francis Guthrie and Augustus De Morgan were among the first to consider the question, spurred by the coloring of maps.

Subsequently, in 1968, Ringel and Youngs tried to prove the formula and came to the conclusion that in a Ringel and Youngs (1968) proved that for genus g>0, the upper bound provided by the Heawood conjecture also give the necessary number of colors, with the exception of the Klein bottle (for which the Heawood formula gives seven, but the correct bound is six).

The theorem gained significant attention in the 20th century when it became one of the most famous unsolved problems in mathematics. In 1976, Kenneth Appel and Wolfgang Haken announced a proof of the Four Color Theorem. However, their proof was highly controversial because it relied on computer assistance to check an extensive number of cases.

Later Verification

The use of computer assistance raised concerns about the accessibility and verifiability of the proof. Over the years, mathematicians worked on simplifying and verifying the proof. The proof’s complexity made it challenging for a complete manual verification.

Mathematicians later provided simplified and more accessible proofs of the Four Color Theorem, but these still relied on computer-aided techniques. The use of computers in the proof generated discussions about the nature of “proofs” in mathematics.

While the Four Color Theorem has been proven, the nature of its proof has sparked debates about the role of computers in mathematical proofs and the level of confidence one can have in computer-assisted proofs. The theorem remains a significant historical and mathematical landmark, showcasing the intricate connections between mathematics and real-world problems.

So What Does the Four Colour Theorem Look Like? 

A map that only consists of four colors would look something like the following:

four color theorem map

four color theorem map

Credits: https://plus.maths.org/content/myths-maths-four-colour-theorem

However, despite its truth, as Professor Chris Budd accurately points out, “the four colour theorem as carefully stated (for non-contiguous planar graphs) is certainly true. But one thing it does not necessarily apply to is an actual map.”

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