The Four-Color Theorem

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The four-color theorem is a fascinating mathematical discovery. At first glance, it seems like a simple statement: any map, no matter what shapes it is composed of or how complex it is, can be colored using no more than four distinct colors in such a way that no two adjacent regions share the same color. Yet, beneath this straightforward claim, there lies rich mathematical history, controversy, and ingenuity. From its origins as a conjecture in 1852 to its revolutionary computer proof over a century later, the Four-Color theorem has challenged mathematicians by pushing the boundaries for proof methods and providing applications beyond cartography. This webpage will take you through the past of the Four-Color theorem, explain the mathematics that make this possible, and show the profound impacts on fields as diverse as network design and ecology. Click on the links in the table of contents to learn more about the history, mathematics, and applications of the four-color theorem.
The Four-Color Theorem: Every planar graph or, equivalently, every map on a plane or sphere can be colored using no more than four colors, such that no two adjacent regions (regions sharing a common boundary segment, not just a point) have the same color (Robertson et al., 2017).

Use the slider corresponding to each polygon to change the color of the polygon. See if you can create a map where no two regions with a common boundary share the same color.

Questions to think about:
1. How can you color this map so that no two adjacent regions share the same color?
2. Is it possible to color the map using fewer than four colors? Why or why not?
3. Why in the real world might it be useful to color adjacent regions differently, using as few of colors as possible?

Another applet to play around with different kinds of maps:

you can also use this link to create your own map and color it according to the 4-color theorem: Four-Color Map Theorem- Try it Yourself


The four-color theorem is unique in that it was proved largely with the help of a computer. Watch these videos for an overview of the theorem, the history, and the significance of a computer proof.

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