graph coloring section 6.4. 6.4 graph coloring 2 color the counties in this map (white is not a...
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Graph Coloring
Section 6.4
6.4 Graph Coloring 2
Color the counties in this map (White is not a color)
6.4 Graph Coloring 3
How many colors did you use?
1. 2 - 3
2. 4 - 5
3. 6 - 7
4. 8 - 9
6.4 Graph Coloring 4
6.4 Graph Coloring 5
How many colors did you use?
1. 2 - 3
2. 4 - 5
3. 6 - 7
4. 8 - 9
6.4 Graph Coloring 6
Two rules for coloring
1.
2.
6.4 Graph Coloring 7
Color each map
6.4 Graph Coloring 8
Draw a map that requires 5 colors
Example 5 colors
6.4 Graph Coloring 9
The Color Theorem
Any map can be colored with . or
fewer colors
Guthrie et al
6.4 Graph Coloring 10
Color The Vertices
A
G
F C
E D
BA
GF
C
E
DB
Graph Theory Lesson 8
6.4 Graph Coloring 11
You do one
WA
NV
OR
TX
UT
NM
COAZ
CA
WA
OR
NVUT
NM
CO
AZ
CA
TX
6.4 Graph Coloring 12
Geometry and Graphs
1. If a graph contains a , the graph requires at least 3 colors
2. If a graph contains a , the graph requires at least 4 colors
3. If a graph contains a , the graph requires at least 5 colors
Cartoon
6.4 Graph Coloring 13
“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}
6.4 Graph Coloring 14
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}
6.4 Graph Coloring 15
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}
6.4 Graph Coloring 16
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
6.4 Graph Coloring 17
“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
6.4 Graph Coloring 18
A: English, Science, Politics D: English, French, Art
B: Science, Politics, Philosophy E: Politics, Art, Philosophy
C: Math, Philosophy, Art F: Math, Science, Music
6.4 Graph Coloring 19
How many days are required?
1. 3
2. 4
3. 5
4. 6
6.4 Graph Coloring 20
A: English, Science, Politics D: English, French, Art
B: Science, Politics, Philosophy E: Politics, Art, Philosophy
C: Math, Philosophy, Art F: Math, Science, Music
G: Music, Philosophy H: Science, French
6.4 Graph Coloring 21
If G is added how many days are required?
1. 3
2. 4
3. 5
4. 6
6.4 Graph Coloring 22
If both G and H are added how many days are required?
1. 3
2. 4
3. 5
4. 6
6.4 Graph Coloring 23
“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?
6.4 Graph Coloring 24
• 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
6.4 Graph Coloring 25
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
6.4 Graph Coloring 26
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
6.4 Graph Coloring 28
Francis Guthrie1831-1899
6.4 Graph Coloring 29
Wolfgang Haken and Kenneth Appel
6.4 Graph Coloring 30
Scheduling Conflicts
Oh, What to do? What to
Dooooo?
6.4 Graph Coloring 31
Ex. 1 Contradiction to the Four-Color Problem?
3
213
21 3
21
3
21
?
Solution
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