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# a7 - 1 i h 9 8 g 7 d 4 e 5 f 6 c 3 b 2 a Let A = 1 2 3 4 5...

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MATH 239 Assignment 7 This assignment is for practice only, and is not to be handed in. 1. Let G be a graph with 2 n vertices such that every vertex has degree at least n . Prove that G has a perfect matching. (Hint: any non-perfect matching admits an augmenting path.) 2. Give an example of a 3-regular graph that does not have a pefect matching. (Note that such a graph cannot be bipartite.) 3. Let G be a bipartite graph with vertex classes A and B , where | A | = | B | = 2 n . Suppose that | N ( X ) | ≥ | X | for all subsets X A with | X | ≤ n , and | N ( X ) | ≥ | X | for all subsets X B with | X | ≤ n . Prove that G has a perfect matching. 4. Apply the bipartite matching algorithm to find a maximum matching and a minimum cover of the following graph:
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Unformatted text preview: 1 i h 9 8 g 7 d 4 e 5 f 6 c 3 b 2 a Let A = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } , and let B = { a,b,c,d,e,f,g,h,i } . The thick edges are the matching edges. 5. Let G be a bipartite graph with bipartition A,B such that | A | = | B | , and for every nonempty subset D ± A , we have that | N ( D ) | > | D | . Prove that, for every edge e ∈ E ( G ), there is a perfect matching containing e . 6. Let K is a matching in a nonbipartite graph G . Show that if K is not maximum, then it admits an augmenting path. (Hint: for any larger matching L , consider only the edges that are in K or L , but not both.)...
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