assgt9 - A | = | B | Suppose that we have | N D |> | D |...

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MATH 239 ASSIGNMENT 9 NOT TO BE HANDED IN 1. Let G be a bipartite graph with bipartition A,B , and a maximum matching M that does not saturate all vertices in A . Prove that | N ( X ) | < | X | , where X A is formed by the XY -construction described in Section 7.2. 2. In the bipartite graph G drawn below, use the bipartite matching algorithm starting with the matching consisting of the thick edges, to determine a set D A with | N ( D ) | < | D | . 1 2 3 4 5 a b c d e A B G 3. Let G be a bipartite graph with bipartition A,B such that |
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Unformatted text preview: A | = | B | . Suppose that we have | N ( D ) | > | D | for every proper nonempty subset D of A . Prove that for every edge e of G , there is a perfect matching containing e . 4. A deck of 52 playing cards is arranged in a rectangular grid of 4 rows and 13 columns. Prove that there is a set of 13 cards, all of diFerent values and in diFerent columns. 5. Determine a minimum cover for the n-cube, for each n ≥ 0....
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