The Four-Color Theorem
1 The map-coloring problem
Given a map (of the continental USA, or the counties of England, or the school districts of Kansas, or. . . ),
you want to color each of the regions so that adjacent regions never receive the same color. For example, the
map on the left is colored correctly. The map on the right is not correct, because Nevada and Idaho, which
share a little bit of border, are both colored green.
WRONG
CO
NV
OR
CA
UT
NMAZ
CO
NV
OR
WA
CA
UT
NMAZ
ID
WA
ID
How many colors are necessary?
Of course, the answer to this question depends on the particular map. So what the problem is really
asking is this: Is there some number of colors that is always enough, no matter what the map looks like?
We have to be a little more precise. First, by “adjacent” we mean “sharing a common piece of border”,
not just “meeting at a point”. (For example, in the maps above, it is okay that Utah and New Mexico are
both colored blue.) Otherwise, it would be possible to construct maps that required arbitrarily high numbers
of colors, so the problem would make no sense.
Second, we have to assume that each region is contiguous — for example, the Lower and Upper Peninsulas
of Michigan are not part of the same region, and can have different colors. Again, if we allow non-contiguous
regions then the problem would not have a well-defined answer.
Even with these caveats, it’s not too hard to construct maps for which at least four colors are required.
The map above is an example. Here’s why.
Each two of the states California, Nevada and Arizona share a common border, so they need three
different colors — say blue, green and red respectively, as in the left-hand figure. If we don’t want to use a
fourth color, then Utah has to be blue (because it has a red neighbor and a green neighbor) and then Idaho
has to be red (because it has a blue neighbor and a green neighbor). But then we’re stuck when we try to
color Oregon: it has a neighbor of each color, so we have no choice but to introduce a fourth color.
2 A reformulation using graph theory
Remember that a graph consists of a set of vertices and a set of edges, where each edge joins two of the
vertices. In the map-coloring problem, we can define a graph whose vertices are the regions of thre map, and
whose edges are the pairs of regions sharing a common border. For the map above, the graph has vertices
1
