370 16. [BB] Three exam periods are required since this is the chromatic number of the graph. 17. We draw a graph whose vertices correspond to courses and an edge means that the two corresponding exams should not be scheduled at the same time (because there are stu-dents taking each course). The chromatic number of the graph is 4 (a 4-coloring is shown). This is best possible be-cause the graph contains K4 (E, M, F, P). The minimum number of exam periods to avoid conflicts is four. 3 R 4 5 W R Solutions to Exercises 6 B WE~MR pY F G y B C R 18. Our proof does not work for four colors. We show why by analyzing our proof of The Five Color Theorem. Let v be a vertex of degree five, adjacent to vertices Vb V2, V3, V4, V5, and let go = g " v. The induction hypothesis would say that X(Qo) ::; 4. Hence two of the Vi would be assigned the same color. Suppose the colors R, W, B, G were assigned as in the figure and also that VIV5 and V2V4 were edges (as shown). Here there is no path which alternates colors between VI and V3, but
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