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
This is the end of the preview. Sign up
access the rest of the document.