392 Solutions to Exercises (b) After the first night, we remove the edges corresponding to the dates which occurred. We are left with a bipartite graph in which every vertex has degree 4. Again, Problem 6 guarantees a perfect matching. Continue for the five nights, reducing the degree by 1 each time. 6. (a) [BB] Bruce +-t Maurice, Edgar +-t Michael, Eric +-t Roland, Herb +-t Richard. (b) If Roland and Bruce share a canoe, then Maurice and Michael would both have to be with Edgar. (c) No. If Bruce and Roland were together, then both Maurice and Michael would have to be in the same canoe with Edgar. On the other hand, if Bruce and Roland were not in the same canoe, then Bruce and Roland would have to be in the same canoe with Maurice and Michael (in some order), leaving no one for Edgar since he refuses to be with Herb. 7. (a) [BB] Construct a bipartite graph with vertex sets VI and V2, where VI has n vertices corresponding to AI, ... , An, V2 has one vertex for each element of S and there is an edge joining Ai to s if and only if s E Ai.
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