There is more than one way to color a MPG so that it satisfies the four-color theorem. For example, the following table lists 24 different ways to color the graph shown below. Note that R = red, B = blue, C = green, and Y = yellow.
The graphs of this chapter include dashed lines to help show you that these are MPG’s. Imagine moving the dashed lines “outside.”
Note that these 24 ways are really just variations of a single way (ordered ABCDE). If you swap blue and green, for example, BGRGY becomes GBRBY, and then if you swap blue and yellow GBRBY becomes GYRYB.
We can get all 24 ways by color swapping.
We can reduce the number 24 down to 1 if we ask a slightly different question. Instead of asking, “How many ways are there to color the graph?”
we can ask, “After fixing the colors of one triangle, how many ways are there to color the remaining vertices?” You don’t want to first color three random vertices because if it turns out that two of those colors needed to be the same in order to four-color the graph, you’ll run into a problem. By first coloring three vertices that lie on one triangle, you “know” that those vertices must be different colors.
Looking at triangle ABC, we could choose A to be red, B to be blue, and C to be green. It then follows that D is blue and E is yellow. This results in the single answer RBGBY.
Now let’s compare how many ways there are to color two different MPG’s with 6 vertices such that they satisfy the four-color theorem. The left graph below can be colored 24 ways, and the right graph below can be colored 96 ways. The dashed lines can be moved “outside.”
Rather than list dozens of ways to color each graph, as we did with the previous graph, let’s alter the question in order to reduce the total number of ways. Let’s choose to color A red, B blue, and G green, which we may do because in both cases ABC happens to be a triangle (which guarantees that these three vertices will have different colors). After coloring A red, B blue, and G green, there is only one way to color the left graph (RBGYBG), whereas there are four ways to color the right graph (RBGRBG, RBGRBY, RBGRYG, and RBGYBG).
• In the left graph, since D connects to A, B, and C, we must color D yellow once A, B, and C are set. Similarly, since F connects to A, B, and D, we must color F green once A, B, and D are set. The only color
remaining for E is blue.
• The right graph is actually three-colorable (RBGRBG). Here, D can be the same as A (red) or it can be yellow, E can be the same as B (blue) or it can be yellow, and F can be the same as C (green) or it can be yellow.
However, since D, E, and F connect to one another, only one of these can be yellow. This gives four possible colorings.
The left graph is more restrictive because A, B, C and D form a K4 subgraph and because A, B, D, and F form another K4 subgraph. The right graph is less restrictive because it doesn’t have any K4 subgraphs. A K4 graph is a complete graph with 4 vertices (see the next page).
In the K4 graph, every vertex connects to all three of the other vertices. The first graph of this chapter (with 5 vertices) has K4 subgraphs (like ABCE) and one of the graphs with 6 vertices has K4 subgraphs (such as ABCD). These two graphs could be colored 24 ways; or after first coloring A, B, and C, there was one way to color the remaining vertices. The other graph with 6 vertices doesn’t have any K4 subgraphs. This graph could be colored 96 ways; or after first coloring A, B, and C, there are still four ways to color the remaining vertices.
Let’s examine some more MPG’s and see if K4 plays a similar role in restricting the number of ways that the graphs can be colored and satisfy the four-color theorem. The following MPG’s have 7 vertices. As usual, the dashed lines help you visualize which of the edges can be moved “outside” of the polygon to convince you that these are MPG’s. Each of these has triangle ABC, so we may choose to color A red, B blue, C green, and count how many ways there are to color the remaining vertices such that the coloring satisfies the four-color theorem.
• The left MPG can be colored one way: RBGYGBG (ordered A thru G, as usual).
• The second MPG from the left can be colored one way: RBGBYGY.
• The third MPG from the left can be colored four ways: RBGRBGY, RBGRBYG, RBGRYBG, and RBGYBYG.
• The right MPG can be colored five ways: RBGRBYG, RBGRYBG, RBGYBYG, RBGYRBG, and RBGYRYG.
Now let’s compare the number of ways that each of the previous MPG’s can be colored with its K4 subgraphs. (We’re looking for subgraphs that can be found without making any subdivisions and without contracting any edges.)
• The left MPG can be colored one way. Its K4 subgraphs are ABCD, ABDG, ADEF, and ADFG. All 7 vertices participate in K4 subgraphs.
• The second MPG from the left can be colored one way. Its K4 subgraphs are ABCG, ACDG, ADEF, and ADFG. All 7 vertices participate in K4 subgraphs.
• The third MPG from the left can be colored four ways. Its only K4 subgraph is AEFG. Only 4 of its vertices participate in K4 subgraphs.
• The right MPG can be colored five ways. It doesn’t have any K4 subgraphs. None of its vertices participate in K4 subgraphs.
The following graphs show a half dozen examples of the many MPG’s that can be drawn with 12 vertices. These are among the more extreme examples in terms of restrictiveness. Imagine moving the dashed lines “outside” of the polygon to see that these are MPG’s.
Not all of the previous graphs share the same triangles, so we will set the first three colors differently for each case.
• For the top left MPG, we will first color B blue, G green, and L yellow.
The graph can then be colored one way: A red, C yellow, D blue, E yellow, F blue, K blue, J yellow, I blue, and H yellow. This gives the coloring RBYBYBGYBYBY (from A thru L, as usual). Its K4 subgraphs are
ABCG, ABGL, ACDG, ADEG, AEFG, AGHI, AGIJ, AGJK, and AGKL.
All 12 vertices participate in K4 subgraphs.
• For the top center MPG, we will first color A red, D blue, and G green.
The graph can then be colored one way: C yellow, B blue, J yellow, K blue, L yellow, F red, E yellow, H red, and I blue. This gives the coloring RBYBYRGRBYBY. Its K4 subgraphs are ABCG, ACDG, ADGJ, AGJK, AGKL, DEFG, and GHIJ. All 12 vertices participate in K4 subgraphs.
• For the top right MPG, we will first color A red, C blue, and G green.
The graph can then be colored one way: K yellow, B green, L blue, E yellow, F red, D green, I blue, J red, and H yellow. This gives the coloring RGBGYRGYBRYB. Its K4 subgraphs are ABKL, ACEG, AGIK, CDEF, GHIJ, and GIJK. All 12 of its vertices participate in K4 subgraphs.
• For the bottom left MPG, we will first color A red, B blue, and L yellow.
There are many ways to color this graph, such as RBYBGYGRYRGY or RBYBRYBGYGRY. It doesn’t have any K4 subgraphs.
• For the bottom center MPG, we will first color A red, E blue, and I yellow. There are many ways to color this graph, such as
RBYGBYGRYBGY or RBRYBRGBYRBG. It doesn’t have any K4 subgraphs.
• For the bottom right MPG (which is noteworthy in that all 12 vertices are degree 5), we will first color A red, B blue, and I green. There are many ways to color this graph, such as RBRYGBGYGBRY or
RBYGYBGRGYRB. It doesn’t have any K4 subgraphs.
What trends have you noticed in this chapter?
Here is one trend that you may have noticed: A MPG that contains a K4 subgraph (without making any subdivisions or contractions) tends to have fewer ways to be colored (using no more than four colors) compared to a MPG (with the same number of vertices) that doesn’t contain any K4 subgraphs.
This suggests that we might be able to add edges to less restrictive MPG’s in order to make them more restrictive. Of course, if you add an edge to a MPG, it won’t be a MPG anymore, but just like the concept of triangulation, extra edges don’t hurt. If we can add an extra edge in such a way that it is easier to show that the new graph is four-colorable than it was to show for the old graph, adding an extra edge may be helpful.
In particular, it seems that it may be helpful to add an edge to a less restrictive MPG in a way that creates K4 subgraphs. We will explore this idea in Chapter 15. As we will see in later chapters, K4’s can be useful in a few different ways:
• Their presence helps make coloring a MPG less restrictive.
• K4’s often include a vertex with degree 3. In Chapter 11, we’ll see that some graphs with K4’s can be trivially colored by removing vertices with degree 3.
• Every K4 contains a separating triangle (Chapter 12). In Chapters 12-13, we’ll explore how such a separating triangle can be used to divide a MPG into smaller MPG’s.
If we add an extra edge to a MPG to create a K4 subgraph, we may wish to take advantage of these properties of K4’s. We will explore adding edges in Chapter 15, removing edges in Chapter 17, and the full significance of adding or removing edges in Chapter 27.
CHAPTER 9 EXERCISES
1. Each MPG below has three vertices already colored. For each MPG:
• Color the remaining vertices so that it satisfies the four-color theorem.
• Determine how many different ways there are to color the remaining vertices.
• Determine how many K4 subgraphs there are (without making any subdivisions or contractions). List the K4 subgraphs. How many vertices participate in K4’s?
Challenge problem 1: We discussed two different methods for counting the number of ways that a MPG can be four-colored. One method is simply to count how many ways a MPG can be four-colored. A second method is to first set the colors of the vertices of one face of the MPG and then determine how many ways the MPG can be four-colored from this point onward. For this second method, does it matter which face is colored first? Either provide an example which demonstrates that this does matter or prove that it doesn’t matter. How does the answer for the second method relate to the answer for the first method (if at all)?
Challenge problem 2: MPG’s that contain K4 subgraphs (without making any subdivisions or contracting edges) are either trivially four-colorable (as we’ll explore in Chapter 11) or can be reduced to MPG’s with fewer vertices (as we’ll discuss in Chapters 13 and 16). When a MPG contains a K4 subgraph, it either has a vertex with degree three that may be removed from the graph or has a separating triangle that lets us split the MPG into two smaller MPG’s (with the K4 no longer present), so that if we could prove that all MPG’s that don’t contain K4 subgraphs are four-colorable, we could easily prove the four-color theorem. (We’ll explore these ideas in later chapters.) You may have observed that MPG’s that don’t contain K4 subgraphs tend to be four-colorable more than one way. If you could prove that every MPG that doesn’t contain a K4 subgraph is four-colorable more than one way (or if you could prove that it is four-colorable at least one way), then you could prove the four-color theorem. Either use this idea to prove the four-color theorem or explain why it is impossible or very challenging to prove that every MPG that doesn’t contain a K4 subgraph is four-colorable at least one way.
Note: The answer key doesn’t include answers to the challenge problems.
These problems are intended to encourage you to think about the ideas.
However, you may wish to consider how the second challenge problem relates to Chapter 27.
Challenge problem 3: The left MPG below has one full B-Y Kempe chain and one full G-R Kempe chain. If instead the left graph is divided into B-G and R-Y Kempe chains or divided into B-R and G-Y Kempe chains, will each chain be full or will any of the chains be separated like the G-R Kempe chains shown in the right graph below? If all possible Kempe chains in the left graph are full, can you prove that if two complementary Kempe chains (like B-Y and G-R) are full in any MPG that all possible Kempe chains will also be full?
Does the number of ways that a MPG is four-colorable (without fixing any of the colors first) relate to the number of full or separated Kempe chains? If so, how? (For example, is every MPG with two complementary Kempe chains that are both full four-colorable 24 ways, and do these 24 ways correspond to 24 independent color swaps, like interchanging R and B? If there are any separated Kempe chains, does each separated segment increase the number of possible color swaps, thereby increasing the number of ways that the MPG can be colored?)
Note: The answer key doesn’t include answers to the challenge problems.
These problems are intended to encourage you to think about the ideas.
However, once you read Chapter 22, you may wish to review your solution to this problem.