Graph Coloring Section 6.4. 6.4 Graph Coloring 2 Color the counties in this map (White is not a color)

Graph Coloring

Section 6.4

6.4 Graph Coloring 2

Color the counties in this map (White is not a color)


How many colors did you use?

1. 2 - 3

2. 4 - 5

3. 6 - 7

4. 8 - 9



How many colors did you use?

1. 2 - 3

2. 4 - 5

3. 6 - 7

4. 8 - 9


Two rules for coloring

1.

2.


Color each map


Draw a map that requires 5 colors

Example 5 colors


The Color Theorem

Any map can be colored with . or

fewer colors

Guthrie et al


Color The Vertices

A

G

F C

E D

BA

GF

C

E

DB

Graph Theory Lesson 8

http://www.utc.edu/Faculty/Christopher-Mawata/petersen2/lesson8.htm


You do one

WA

NV

OR

TX

UT

NM

COAZ

CA

WA

OR

NVUT

NM

CO

AZ

CA

TX


Geometry and Graphs

1. If a graph contains a , the graph requires at least 3 colors



Cartoon


“Student Council Committees”A student council consists of 7 students a, b, c, d, e, f, g. Each student belongs to several of 6 committees

These committees meet weekly for an hour at the same time (noon). All members must be present before business can be conducted. There are lots of meeting rooms available.

What is the fewest number of days required to schedule all 6 committees?

Executive (E) = {a, b, c} Ways/Means (W) = {b, d, e}

Finance (F) = {a, b, d} By-Laws (B) = {a, c, g}

Social (S) = {e, f} Recruiting (R) = {c, e, f, g}


M T W TH F

What meetings should be scheduled on which days so that no conflicts arise?

Executive (E) = {a, b, c} Ways/Means (W) = {b, d, e}

Finance (F) = {a, b, d} By-Laws (B) = {a, c, g}

Social (S) = {e, f} Recruiting (R) = {c, e, f, g}


Executive (E) = {a, b, c, f, h}

Finance (F) = {a, b, d, e}Social (S) = {c, e, f}Ways and Means (W) = {b, d,

e, g, h}By-Laws (B) = {a, c, g}Recruiting (R) = {c, e, f, g}Sports (SP) = {a, b, c}Coordinating (C) = {a, b, d, g}Academic (A) = {b, e, f, h}Newsletter (N) = {b, d, e, h}Outreach (O) = {a, c, g}Fund-raising (FR) = {a, c, e, f, g}


1. Vertices = Parties .

4-Step Algorithm

Executive (E) = {a, b, c}Finance (F) = {a, b, d}Social (S) = {e, f}Ways/Means (W) = {b, d, e}By-Laws (B) = {a, c, g}Recruiting (R) = {c, e, f,

g}

Exec

Recru

SocBy-Laws

WM

Fin

2. Edges join parties3. Color the vertices

4. # colors = # days required

M T W Th F


“Final Exams”

A school has six graduating seniors: Adams (A), Black (B), Courtois (C), D’Amico (D), Epstein (E), and Flaherty (F)You must prepare a final exam schedule for these students. Students can take only one exam each day

What is the fewest number of days required to schedule all 8 exams?

A: English, Science, Politics D: English, French, Art

B: Science, Politics, Philosophy E: Politics, Art, Philosophy

C: Math, Philosophy, Art F: Math, Science, Music






How many days are required?

1. 3

2. 4

3. 5

4. 6





G: Music, Philosophy H: Science, French


If G is added how many days are required?

1. 3

2. 4

3. 5

4. 6


If both G and H are added how many days are required?

1. 3

2. 4

3. 5

4. 6


“Desperate Housewives”

You owe party invitations to several sets of friends: Browns, Caldwells, Fortins, Grandes, Martens, Nevins, and Princes. Three nights (Thursday, Friday, and Saturday) are available

However, you know that it would make for a happier time for all if several of the couples were not to come on the same night

Can you schedule all of your friends on the three nights?


• Nevins don’t get along with Browns or Fortins

• Martins usually argue politics with Nevins

• Princes and Grandes are in-laws whose kids are

fighting

• Prince just sued Brown and Fortin

• Caldwell and Martin are spiteful business

competitors

• One of the Browns is having an affair with one

of the Caldwells

• Fortin’s owe the Grandes a considerable

amount of money


How many nights are necessary?

1. 2

2. 3

3. 4

• Nevins don’t get along with Browns or Fortins• Martins usually argue politics with Nevins• Princes and Grandes are in-laws whose kids are

fighting• Prince just sued Brown and Fortin• Caldwell and Martin are spiteful business

competitors• One of the Browns is having an affair with one of

the Caldwells• Fortin’s owe the Grandes a considerable amount

of money


One more complication

Nevins don’t get along with Browns or FortinsMartins usually argue politics with NevinsPrinces and Grandes are in-laws whose kids are fightingPrince just sued Brown and FortinCaldwell and Martin are spiteful business competitorsOne of the Browns is having an affair with one of the CaldwellsFortin’s owe the Grandes a considerable amount of moneyBrowns had a long-standing feud with both the Fortins and the Grandes?.

1. 2

2. 3

3. 4

4. 5

End of 6.4


Francis Guthrie1831-1899


Wolfgang Haken and Kenneth Appel


Scheduling Conflicts

Oh, What to do? What to

Dooooo?


Ex. 1 Contradiction to the Four-Color Problem?

3

213

21 3

21

3

21

?

Solution

Meta - Material

Documents

Graph Coloring Section 6.4. 6.4 Graph Coloring 2 Color the counties in this map (White is not a color)