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Coloring Graphs This handout: • Coloring maps and graphs • Chromatic number • Applications of graph coloring Coloring maps • Color a map such that two regions with a common border are assigned different colors. • Each map can be represented by a graph: – Each region of the map is represented by a vertex; – Edges connect two vertices if the regions represented by these vertices have a common border. • The resulting graph is called the dual graph of the map. Coloring Graphs • Definition: A graph has been colored if a color has been assigned to each vertex in such a way that adjacent vertices have different colors. • Definition: The chromatic number of a graph is the smallest number of colors with which it can be colored. In the example above, the chromatic number is 4. Coloring Planar Graphs • Definition: A graph is planar if it can be drawn in a plane without edge-crossings. • The four color theorem: For every planar graph, the chromatic number is ≤ 4. Was posed as a conjecture in the 1850s. Finally proved in 1976 (Appel and Haken) by the aid of computers. An application of graph coloring in scheduling Twelve faculty members in a mathematics department serve on the following committees: Undergraduate education: Sineman, Limitson, Axiomus, Functionini Graduate Education: Graphian, Vectorades, Functionini, Infinitescu Colloquium: Lemmeau, Randomov, Proofizaki Library: Van Sum, Sineman, Lemmeau Staffing: Graphian, Randomov, Vectorades, Limitson Promotion: Vectorades, Van Sum, Parabolton The committees must all meet during the first week of classes, but there are only three time slots available. Find a schedule that will allow all faculty members to attend the meetings of all committees on which they serve. An application of graph coloring in exam scheduling Suppose that in a particular quarter there are students taking each of the following combinations of courses: Math, English, Biology, Chemistry Math, English, Computer Science, Geography Biology, Psychology, Geography, Spanish Biology, Computer Science, History, French English, Psychology, Computer Science, History Psychology, Chemistry, Computer Science, French Psychology, Geography, History, Spanish What is the minimum number of examination periods required for the exams in the ten courses specified so that students taking any of the given combinations of courses have no conflicts? Find a schedule that uses this minimum number of periods.