Graph Coloring

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 X X -- -- --

Marty

X -- -- X X --

Kim -- X -- -- -- X

Lois X -- X -- -- --

Dot X -- -- -- X --

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?

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.

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:

Club Graph

Math Honor

Science Art Pep

Spanish

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.

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.

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.

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

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.

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.

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.

Example• Color the following map using

four of fewer colors:

A

B

D

EC

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 B

C

E

D

(red) (yellow)

(blue)

(green)

Coloring the Graph• The colored graph would look

like this:

A

B

D

C EC B

Practice Problems1. Find the chromatic number for

each of the graphs below:a. b.

c.

Practice Problems (cont’d)2. 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.

Practice Problems (cont’d)3. 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?

4. After having colored the vertex with the most conflict, which other vertices can receive the same color?

5. Which vertex would then get the second color? Which other vertices could get that same second color?

Practice Problems (cont’d)6. Then would the coloring

process be complete?7. What is the chromatic number

of K2? K3? K4? KN?

8. A cycle is a path that begins and ends at the same vertex and does not use any edge or vertex more than once.a. If a cycle has an even number of

vertices, what is its chromatic number?

b. What is the chromatic number of a cycle with an odd number of vertices?

Practice Problems (cont’d)9. 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?

A

B

CD

E

F

G

H

Practice Problems (cont’d)10. 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 5

7 1, 4, 5

Practice Problems (cont’d)How many different storage

facilities are necessary in order to keep all seven chemicals?

11.Color the following map using only three colors.

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)

WA

OR

CA

NV

ID

UT

AZ

NM

CO

WY

MT

OK

KS

NBIA

MO

AK

LAMS

TN

KY

IL