 The map-coloring problem considers a plane divided into a number of regions. Two regions that share a border should be assigned different colors, however, two regions that share a point can be colored the same. So how many colors are required? Mapmakers use as many colors as they need to. Color choices are based on aesthetic principles, as well as political considerations, like having all islands of one state have the same color, or having two parts of a state separated by water have the same color. So a normal person wouldn't worry about how many colors are required. But mathematicians are not normal people. In 1852, Francis Guthrie noted that a map of the counties of England only needed four colors. His brother Frederick brought the question to Augustus de Morgan. Both Guthries had been students of de Morgan and went on to become scientists in their own right. De Morgan brought the problem to the attention of other mathematicians like Hamilton and Cayley. In 1879, Alfred Kemp published a proof of the four-color theorem, and unfortunately it was flawed. However, the essential ideas of Kemp's work became part of our proof of the theorem that any planar graph has a five-coloring. Now the five-coloring in the theorem is actually a vertex coloring, but what about a coloring of map regions? So let's think about that. A graph represents relationships between objects. In a map, two regions are related by whether or not they share a common boundary. So we can represent a plane map as a graph by representing each region by points, joining two points if the regions share a common boundary. Since there are other ways we could produce a graph from a graph, let's call this a derived graph for now. And if g is our graph, we'll use the notation g' for our derived graph. And it's very important to remember, notation is local. Always check to make sure a symbol means what you think it does. So we'll produce this derived graph as follows. For every face in g, create a vertex v' in g' and if two faces are adjacent, place an edge between their corresponding vertices. So let's use the described approach to create the derived graph of the graph shown. The graph has four faces, which in a fit of originality will label one, two, three, and don't forget the outside face, four. So our derived graph will have four vertices, which will label one, two, three, and four. So let's join vertices that share a common edge. So region one shares a boundary with regions two, three, and four. So there are edges between these vertices from one to two and three and four. Likewise, regions two, three, and four share a boundary with the other three regions. So we'll include edges between them. So the derived graph is k4. And it's worth noting the original graph is also k4. Now a useful thing to do in mathematics is to look for problems, not solutions. And in this case, if you're looking for a research question and you should be, this problem suggests several others. For what graphs g is g' equal to g, are there graphs for which the nth derived graph is the original graph? You could also ask questions like, is there any situation where this might be useful? But those who ask and answer these questions are often burdened with money and job offers. We should produce the derived graph as we did, since it's based on how we defined it. But a more intuitive way to produce a derived graph, put a vertex in each face, if two faces are adjacent, join the vertices with an edge passing through the common boundary exactly once. This also produces a derived graph. But is it the same derived graph? It is, but we have to prove it. Well, actually, you have to prove it. So let's use the described procedure to create the derived graph of the graph shown. So we'll place a vertex inside each face and one on the outside region. Then we'll join the vertices across their common edge. And note that the common edge to the outside region can be bent a little bit so we don't cross edges. So any planar graph can be vertex colored using five colors. Since every map can be transformed into a graph, then every map can be colored using five colors, provided the map can be converted into a planar graph. But can it be? We need to show that any planar graph has a derived graph that is also a planar graph. Suppose we superimpose our derived graph and our original graph. So let's take a function which maps points in the plane to the vertices of the derived graph or to zero if the point is on a boundary between two regions. In other words, this function will identify which region a point is in. Suppose our derived graph is not planar, and some edge, we'll call it 1, 2, crosses another edge. Every point on this edge will be assigned by our function to 1 or 2, except for the single point of the boundary between the two regions, which will be assigned 0. If 1, 2 crosses another edge, the incident vertices must be different. So we may assume the other edge is say 3, 4. Again, every point on the edge between vertices 3 and 4 will be assigned by our function to either 3 or 4, except for the single point of the boundary between the two regions, which will be assigned 0. So if two edges intersect, our function has to assign the intersection point a value of 0. This means it corresponds to a point in the original graph that is on the edge separating faces 1 and 2, and also on the edge separating faces 3 and 4. So the two edges in the original graph must also intersect. But since the graph is planar, edges don't intersect. So edges in the derived graph can't intersect either. Consequently, the derived graph of a planar graph is also a planar graph. There's one last step. The derived graph has a five-coloring of the vertices, but does this mean that the corresponding regions could be colored this way? Well, suppose two vertices have the same color. Ahem, same color. Then, there can be no edge between them. Consequently, the corresponding regions do not share a boundary, so they can be assigned the same color. Consequently, any map can be expressed as a planar graph with vertices corresponding to the map regions. Any planar graph has a five-coloring of the vertices, and the same coloring can be applied to the corresponding map regions. So any map can be colored with five or fewer colors. The proceeding suggests that there are actually two related four-color conjectures. Given a planar graph, can we color the faces using just four colors, or can we color the vertices using just four colors? They're related, but they aren't the same problem. Or are they? Let's take a closer look.