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MC 302 Graph Theory Tuesday, 10

Characterizing Bipartite Graphs. Theorem. Let G be a graph with at least 2 vertices. The following statements about G are equivalent:1. G is bipartite.2. G can be properly 2-colored.3. G has no odd cycles.Proof: Equivalence of 1 and 2, and 2 ? 3 were done on worksheet. We'll show 3 ? 2.. Applica

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MC 302 Graph Theory Tuesday, 10

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    1. MC 302 Graph Theory Tuesday, 10/19/04 Talk announcement: Applications of Graph Theory! Wednesday, 5:30/5:45 Refreshments/Talk Harder 202 Extra credit (small!) for attending! Todays reading & exercises: Reading: 3.3-3.4 Exercises: 3.3 #3,5; 3.4 #2,11,14,18. In #11, start with a matching with one edge and then use the Hungarian Algorithm. Today: (Questions?) Worksheet: Bipartite Graphs Application: Matchings and Job Assignments

    2. Characterizing Bipartite Graphs Theorem. Let G be a graph with at least 2 vertices. The following statements about G are equivalent: 1. G is bipartite. 2. G can be properly 2-colored. 3. G has no odd cycles. Proof: Equivalence of 1 and 2, and 2 ? 3 were done on worksheet. Well show 3 ? 2.

    3. Applications of Bipartite Graphs Personnel Assignment Problem A company has workers X1, , Xm and jobs Y1, , Yn. Each worker is qualified to do some jobs, but not others. Can every worker be assigned a job? Optimal Assignment Problem Same basic setup as above, but now each pair (Xi, Yj) is given a weighting wij indicated the effectiveness (e.g. profit to company) of assigning worker Xi to job Yj. How should jobs be assigned to maximize the total effectiveness of the assignments? Marriage Problem There are k men and m women (say, on a reality show), and each male-female pair has expressed whether or not they are willing to marry. How can we pair them up so that all the men are paired with acceptable mates (or the gender-reversed question)?

    4. Matchings All three problems involve forming a matching in a bipartite graph: Definition: Let G be a graph with {V1, V2}. A matching in G is a set of edges, no two of which share an endpoint.

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