1. Graph Coloring

2. Central High School
Below is an organizational table of the students who hold offices in the clubs at Central High School.
Math Club
Honor Club
Science Club
Art Club
Pep Club
Spanish Club
Matt X --
Marty
Kim
Lois
Dot

3. Club Meetings
If each club at Central High wants to meet once a week, since several members hold offices in more than one organization, the schedules will have to be arranged so that the meeting days are scheduled for several days.
Is it possible to come up with such a schedule?
What is the minimum number of days needed for the schedule?

4. Solving the Problem
One way to solve this problem would be to use 5 days for the scheduling.
Notice that the Math and Spanish Clubs meet of Monday and the remaining clubs could meet on the other 4 days.
If the problem is to schedule the meetings in the fewest number of days, then this solution is not optimal.
It is possible to create a schedule using only 3 days.

5. Trial by Error
Finding that schedule by trial and error for this problem would not be too difficult, but a mathematical model would be helpful for more complicated problems.
We could construct a graph in which the vertices represent clubs at Central High School and the edges indicate that the clubs share an officer and so can not meet on the same night.
Such a graph would look like:

6. Club Graph
Honor
Math
Spanish
Science
Art
Pep

7. Labeling the Days
Now we can begin by labeling the graph with the days of the week that the clubs can meet on.
Adjacent vertices must have different labels, since this is where the conflicts occur.
One ways of assigning days to begin with the Math Club and labeling it Monday.
Since no one belongs to both the Math Club and the Spanish Club, the Spanish Club can also be labeled with Monday.

8. Labeling the Days (cont’d)
We will also label the Honor Club with Tuesday.
The Pep Club or the Art Club, but not both, can receive a Tuesday label.
The other label is placed with Wednesday.
The Science Club can also receive a Wednesday label.
The resulting schedule is an optimal solution to the problem, but notice that it is not unique.

9. Coloring Problems Problems of this type are called coloring problems.
They are called this because historically the labels placed on the vertices of the graphs were called colors.
The process of labeling the graph is referred to as coloring the graph.
The minimum number of labels or colors that can be used is known as the chromatic number of the graph.
The chromatic number for the graph we just completed is three.

10. The Four-Color Conjecture
This type of problem attracted the attention of several 19th century mathematicians such as Augustus de Morgan, William Rowan Hamilton
and Arthur Cayley.
They became interested in the problem because of the four-color conjecture.
This conjecture stated that any map
that could be drawn on the surface
of a sphere could be colored with,
at most, four colors.
Augustus de Morgan
Arthur Cayley

11. Four-Color Conjecture
This problem intrigued mathematicians for over 100 years.
During that time many tried prove the conjecture but flaws were always found in the proofs.
It wasn’t until 1976 that Kenneth Appel and Wolfgang Haken of U of Illinois actually solved the problem, that the four-color conjecture became the four-color theorem.
They proved this theorem in a very different way than all the other proofs had been attempted.

12. Appel and Haken’s Proof
They proved their theorem using a high-speed computer a first for the field of mathematics.
When the proof was completed, they had used over 1,200 hours of time on three different computers.
The honor them the University of Illinois had a postage meter stamp created.

13. Map Coloring
One way to approach map coloring is to represent each region of the map with a vertex of a graph.
Two vertices are connected with an edge if the regions they represent have a common border.
Coloring the map is then the same process as coloring the vertices of a graph so that adjacent vertices have different colors.

14. Example
Color the following map using four of fewer colors: A D C E B

15. Finding the Solution
To find a solution, represent the map with a graph in which each vertex represents a region of the map, and draw edges between vertices if the regions on the map have a common border. Then label the graph with a minimum number of colors. A
(red) B
(yellow) E C
(green)
(blue) D

16. Coloring the Graph The colored graph would look like this: A D C B C E

17. Practice Problems
Find the chromatic number for each of the graphs below:
a. b. c.

18. Practice Problems (cont’d)
a. Draw a graph that has four vertices and a chromatic number of three.
b. Draw a graph that has four vertices and a chromatic number of one.
As the number of vertices in a graph increases, a systematic method of labeling the vertices becomes necessary. One way to do this is to create a coloring algorithm.

19. Practice Problems (cont’d)
One way to begin the coloring process is first color the vertices with the most conflict. How can the vertices be ranked from those with the most to those with the least conflict?
After having colored the vertex with the most conflict, which other vertices can receive the same color?
Which vertex would then get the second color? Which other vertices could get that same second color?

20. Practice Problems (cont’d)
Then would the coloring process be complete?
What is the chromatic number of K2? K3? K4? KN?
A cycle is a path that begins and ends at the same vertex and does not use any edge or vertex more than once.
If a cycle has an even number of vertices, what is its chromatic number?
What is the chromatic number of a cycle with an odd number of vertices?

21. Practice Problems (cont’d)
Mrs. Suzuki is planning to take her history class to the art museum. Below is a graph showing those students who are not compatible. Assuming that the seating capacity of the cars is not a problem, what is the minimum number of cars necessary to take the students to the museum? B C D A H G E F

22. Practice Problems (cont’d)
Below is a list of chemicals and the chemicals with which each cannot be stored.
Chemicals
Cannot be Stored With 1
2, 5, 7 2
1, 3, 5 3
2, 4 4
3, 7 5
1, 2, 6, 7 6 7
1, 4, 5

23. Practice Problems (cont’d)
How many different storage facilities are necessary in order to keep all seven chemicals?
Color the following map using only three colors.

24. Practice Problems (cont’d)
12. Draw graphs to represent the maps below. Color the graphs. What is the minimum number of colors needed to color each map? (only the labeled states) IA NB WA IL MT OR KS MO ID KY WY TN NV OK UT AK CO CA MS LA AZ NM