According to the four-color theorem, the vertices of any PG may be colored using no more than four different colors (such as red, blue, green, and yellow) such that [Ref. 1]:
• No two vertices connected by an edge have the same color. (An edge connects two vertices and may be straight, curved, or bent.)
• Every vertex is colored. A single vertex can only be colored using a single color; a multi-colored vertex isn’t allowed.
• The number of vertices is finite. The graph isn’t an infinitely repeated design like a tessellation or fractal. (This particular assumption may not be necessary, but provides a simple starting point with which to approach the four-color theorem.)
• The graph is drawn in the plane or on the surface of a sphere (but other surfaces like a torus are not allowed). Chapter 14 illustrates the concept of the sphere.
• The graph is undirected (the edges don’t have arrows). The graph isn’t disconnected. No edge connects a region to itself (this is called a loop).
There are no double edges.
Recall from Chapter 1 that PG stands for “planar graph.” A PG is a graph that can be drawn in the plane without any crossings. PG’s are special because they can be mapped in the plane.
Note that there are additional requirements for maps. For example, the regions of a map must be contiguous (there can’t be any gaps or lakes between regions). Two regions of a map may be the same color if they meet only at a vertex (and not an edge). For a map, we can’t allow regions to be disjointed (like the United States, for which Alaska and Hawaii are separated from the other 48 states).
As discussed in Chapter 1, a single graph may correspond to a multitude of
different maps, which makes it simpler to analyze a graph. For this reason, this book will focus primarily on graphs from this point forward.
Coloring a graph is no different from coloring a map. Below, we colored both a map (lower figures) and its corresponding graph (upper figures). The numbers 1-4 represent four different colors (such as red, blue, green, and yellow). On the graph, no two regions connected by an edge have the same color. On the map, no two regions that share a border have the same color.
The map and graph on the left side are labeled with letters to help you see how the regions of the map and graph correspond. The numbers on the right side show how they are colored.
We encourage you to attempt to color the graph below using no more than four colors.
Note that it isn’t necessary to actually use colors. You can simply write the
numbers 1, 2, 3, and 4 in each region to represent four different colors (like red, green, blue, and yellow).
You can find the solution to the previous puzzle on the following page. Note that there is more than one possible solution. For example, you could get another solution by swapping all of the 1’s and 3’s, then you could get another solution by swapping all of the 2’s and 3’s of that graph, etc. (Yet there are additional solutions possible besides color swapping.)
After you attempt to color the graph (or any map or graph), check your answers carefully. It is really easy to make a mistake. The following approach can help you check your attempt to color a graph so that it satisfies the four-color theorem:
• Find all of the regions of the same color.
• Check these regions one pair at a time.
• For each pair of regions that are the same color, verify that no lines or curves connect these two regions.
• Repeat these steps for each of the remaining colors.
Once a math lover spends enough time drawing a variety of graphs (and perhaps maps), attempting to color the graphs, and attempting to disprove the four-color theorem, it often seems intuitive that a simple and convincing proof of the four-color theorem should exist. (You are highly encouraged to try these things; they are great ways to learn more about the four-color theorem.) However, once enough time is spent attempting to prove the four-color theorem by hand, a variety of challenges tend to become apparent:
• Even for a fairly small number of vertices like 20, there a great many ways to draw graphs that are structurally different, and it isn’t easy to think of every possible way to draw the graph.
• As the number of vertices increases, the number of ways to draw graphs that differ in structure grows tremendously.
• Once a graph is drawn, it isn’t always easy to tell at a glance whether or not it is a PG. That is, if there are crossings, it takes some effort to
determine whether or not the crossings are avoidable. Recall that we are defining PG to include any graph that can be drawn in the plane without crossings (even if the graph happens to be drawn in a form that has avoidable crossings). Chapter 6 will discuss how to determine this.
• It is difficult to formulate a proof that covers every conceivable scenario.
Following is one possible solution to the previous puzzle:
Check the solution:
• Find the three regions colored 1. Verify that no edge joins two of these 1’s together.
• Find the three regions colored 2. Verify that no edge joins two of these 2’s together.
• Find the three regions colored 3. Verify that no edge joins two of these 3’s together.
• Find the four regions colored 4. Verify that no edge joins two of these 4’s together.
As of the publication of this book, the only known proof of the four-color theorem involves computer calculations; the four-color theorem has yet to be proven by hand [Ref. 13]. We’ll explore a variety of ways, including a few novel ideas, for approaching the problem of how one might go about proving the four-color theorem by hand throughout this book.
CHAPTER 2 EXERCISES
1. Color each region of the map below so that the coloring satisfies the four-color theorem. Treat the unbounded surrounding area (that is, the area outside of the map) as one of the regions.
2. Color each vertex of the graph below so that the coloring satisfies the four-color theorem.
3. Color each vertex of the graph below so that the coloring satisfies the four-color theorem.
4. Color each vertex of the graph below so that the coloring satisfies the four-color theorem.
3 TRIANGULATION
A graph is triangulated if every face is surrounded by three edges (which may be lines or curves), including the “face” that represents the infinite area
“outside” of the graph.
• The left graph below isn’t triangulated because ACDE is a quadrilateral (four-sided).
• The right graph below isn’t triangulated because the infinite area
“outside” of the graph has five sides (B, C, D, E, and F) instead of three.
• The center graph below is triangulated because every face, including the infinite area outside, has three sides. Its faces are ABC, ACE, AFE, ABF, BCD, CDE, DEF and BDF. Note that BDF is the infinite area outside (but recall from the end of Chapter 1 that any graph can be inverted to make any face correspond to the infinite area outside of the graph).
Any graph that isn’t already triangulated can become triangulated by adding one or more edges to the existing graph. Consider the example below.
In the previous diagram, the left graph isn’t triangulated because ABCF and CDEF each have four sides and because the infinite area outside BGDH also has four sides. If we add edges AC, CE, and BD (this one is curved), every face will be a triangle (including the infinite area outside, which is now BDG).
We will use the term maximal planar graph for any PG that has been triangulated in this sense (including the infinite area outside), and we will abbreviate this MPG. An alternative name that is also common is
“triangulated graph.” Since every face of a MPG is triangular (in a loose sense of the word, since any of its three edges may be curved), it may seem like triangulated graph would be the better choice. However, since the term triangulated graph is sometimes used with other meanings in mind, the term MPG is common in order to help avoid possible confusion. (We are abbreviating MPG since we will use this term frequently.)
The following property makes it very useful to triangulate graphs to turn PG’s into MPG’s. If a graph is colored in such a way that it satisfies the four-color theorem, the same four-coloring will still satisfy the four-four-color theorem if one or more edges are removed from the graph. You can see that in the example above. We first colored the MPG on the right. (It turns out that this MPG can be colored using just three colors, but that is unimportant.) We obtained the graph on the left (which is a PG, not a MPG) by removing three edges from the MPG. You can see that the coloring from the MPG on the right still works for the PG on the left after removing the edges from the graph.
Recall from Chapter 1 that a PG is a graph that can be drawn in the plane without crossings. In contrast, a MPG is a special type of PG in that it is fully triangulated, including the infinite area outside.
Any MPG that is colored in such a way as to satisfy the four-color theorem will still satisfy the four-color theorem if any of its edges are removed from the graph. This important classic property of triangulation is the reason that most attempts to prove the four-color theorem only consider MPG’s. If you can prove that the four-color theorem holds are all MPG’s, you will have proven that it holds for all PG’s.
MPG’s are thus central to the four-color theorem. Note that a single MPG may be drawn more than one way, as illustrated below.
The three MPG’s shown above are all isomorphic. The form of the graph in the middle has crossings, but the forms of the graph on the left and right show that these crossings are avoidable. (Recall from Chapter 1 that graphs are isomorphic if they are structurally equivalent in terms of edge sharing.) The following diagrams show that two MPG’s can have the same number of
vertices, yet be structurally different. The MPG on the left has two vertices of degree 5, two vertices of degree 4, and two vertices of degree 3, whereas the MPG on the right has six vertices of degree 4.
As the number of vertices increases, the number of MPG’s with structural differences increases severalfold. For example, with eight vertices, one graph can have two vertices of degree 7, two vertices of degree 3, and four vertices of degree 4, a graph can have four vertices of degree 5 and four vertices of degree 4, and there are many other graphs between these two extremes. Three of the many possibilities are shown below.
CHAPTER 3 EXERCISES
1. For each graph below:
• Indicate whether the graph is a PG or MPG.
• If the graph is a PG, add edges to the graph to turn it into a MPG.
• Once it is a MPG, identify the face that corresponds to the infinite area outside.
2. Add edges to triangulate the left graph below. Once the graph is triangulated, color the graph so that the coloring satisfies the four-color theorem. Color the right graph below the same way. If the added edges are removed from the MPG, will the coloring still satisfy the four-color theorem?
Does the original PG have fewer, the same, or more ways to color it (so that it satisfies the four-color theorem) compared to the MPG?
3. Draw and label a graph corresponding to the map below. Add edges to triangulate the graph. Now draw and label a map corresponding to the MPG.
4. Are any of the graphs below isomorphic? (Recall that two graphs are isomorphic if the graphs are structurally equivalent in terms of edge-sharing.) If so, which ones?
5. Are any of the graphs below isomorphic? If so, which ones?
6. Looking at any map, what can you look at visually that will tell you whether or not the corresponding graph will be triangulated?