1. Graph Coloring Section 6.4

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

3. 6.4 Graph Coloring 3 How many colors did you use? 1. 2 - 3 2. 4 - 5 3. 6 - 7 4. 8 - 9

4. 6.4 Graph Coloring 4

5. 5 How many colors did you use? 1. 2 - 3 2. 4 - 5 3. 6 - 7 4. 8 - 9

6. 6.4 Graph Coloring 6 Two rules for coloring 1. 2.

7. 6.4 Graph Coloring 7 Color each map

8. 6.4 Graph Coloring 8 Draw a map that requires 5 colors Example 5 colors

9. 6.4 Graph Coloring 9 The Color Theorem Any map can be colored with. or fewer colors Guthrie et al

10. 6.4 Graph Coloring 10 Color The Vertices A G F C E D B A GF C E D B Graph Theory Lesson 8raph Theory Lesson 8

11. 6.4 Graph Coloring 11 You do one WA NV OR TX UT NM CO AZ CA WA OR NV UT NM CO AZ CA TX

12. 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

13. 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}

14. 6.4 Graph Coloring 14 MTWTHF 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}

15. 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}

16. 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 parties 3. Color the vertices 4. # colors = # days required MTW ThF

17. 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, PoliticsD:English, French, Art B:Science, Politics, PhilosophyE:Politics, Art, Philosophy C:Math, Philosophy, ArtF:Math, Science, Music

18. 6.4 Graph Coloring 18 A:English, Science, PoliticsD:English, French, Art B:Science, Politics, PhilosophyE:Politics, Art, Philosophy C:Math, Philosophy, ArtF:Math, Science, Music

19. 6.4 Graph Coloring 19 How many days are required? 1.3 2.4 3.5 4.6

20. 6.4 Graph Coloring 20 A:English, Science, PoliticsD:English, French, Art B:Science, Politics, PhilosophyE:Politics, Art, Philosophy C:Math, Philosophy, ArtF:Math, Science, Music G:Music, PhilosophyH:Science, French

21. 6.4 Graph Coloring 21 If G is added how many days are required? 1.3 2.4 3.5 4.6

22. 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

23. 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?

24. 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

25. 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

26. 6.4 Graph Coloring 26 One more complication 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 Browns had a long-standing feud with both the Fortins and the Grandes?. 1. 2 2. 3 3. 4 4. 5

27. End of 6.4

28. 6.4 Graph Coloring 28 Francis Guthrie 1831-1899

29. 6.4 Graph Coloring 29 Wolfgang Haken and Kenneth Appel

30. 6.4 Graph Coloring 30 Scheduling Conflicts Oh, What to do? What to Dooooo?

31. 6.4 Graph Coloring 31 Ex. 1 Contradiction to the Four-Color Problem? 3 2 1 3 2 1 3 2 1 3 2 1 ? Solution

32. Meta - Material