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From Coloring Maps to Avoiding Conflicts

From Coloring Maps to Avoiding Conflicts. Nathaniel Dean, Robert M. Nehs, and Tong Wu Department of Mathematical Sciences Texas Southern University 3100 Cleburne Avenue Houston, TX 77004. ocean infinite face. countries regions faces.

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From Coloring Maps to Avoiding Conflicts

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  1. From Coloring Mapsto Avoiding Conflicts Nathaniel Dean, Robert M. Nehs, and Tong Wu Department of Mathematical Sciences Texas Southern University 3100 Cleburne Avenue Houston, TX 77004

  2. ocean infinite face countries regions faces Map ColoringCountries with a common boundary must have different colors.

  3. 2 1 4 3 1 1 2 3 3 1 2 1 2 Four Color Problem1852 letter by Augustus de Morgan to Sir William Hamilton: Four colors are required.Do 4 colors suffice? 1976: Appel and Haken proved it using an intricate case analysis on a computer.

  4. Exercise:Draw a map that requires four colors.

  5. 3-Coloring Maps Computer Science project by Malvika Rao (student), McGill U. http://www.cs.mcgill.ca/~rao/cs507/MapColoring.html

  6. The Dual is a Planar Graph.

  7. edge link line 9 vertex node point 7 8 1 5 4 2 3 6 Graph G=(V,E)consists of a set V of objects called vertices and a set E of unordered pairs of vertices.

  8. 2 1 4 2 3 2 1 (G) = 4 Vertex Coloring • A k-coloring is a labeling f:V(G)  {1,2,…,k}. • A k-coloring is proper if xyE(G) implies f(x)  f(y). • G is k-colorable if it has a proper k-coloring. • The chromatic number(G) is the smallest k such that G is k-colorable.

  9. Exercise:Prove  (Moser Graph) = 4.

  10. Party Problem • People P1, P2, …, Pn meet for a party, but certain pairs are incompatible. • Goal: Assign people to rooms so that no two people in the same room are incompatible. • How many rooms are needed?

  11. Solution to the Party Problem Construct a conflict graph G. • V(G) = {P1, P2, …, Pn}. • Pi, PjE(G) iff Pi and Pj are incompatible. • The chromatic number(G) is the least number of rooms. 2 1 4 2 3 2 1 (G) = 4

  12. Scheduling Problem • Five different groups of students {1,2,3}, {6,7}, {1,7,9}, {4,6,8}, {2,3,4} must take exams in the following engineering courses S1, S2, S3, S4, S5, respectively. • Goal: Schedule the exams using a minimum number of time periods.

  13. Solution to the Scheduling Problem Construct a conflict graph G. • V(G) = {S1, S2, S3, S4, S5}. • Si, Sj E(G) iff Si  Sj . • The chromatic number(G) is the minimum number of time periods. {2,3,4} {1,2,3} {4,6,8} {6,7} {1,7,9} (G) = 3

  14. Future Work • Allow student to enter any arbitrary map or (nonplanar) graph. • Add a map library, and allow transferring between map and corresponding graph. • Develop algorithms with student. • Explore applications.

  15. Frank David MAA Lang Andrew Thanks! • Directors:Lang Moore David Smith Frank Wattenberg • Funding Agencies

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