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Unformatted text preview: t 1 or t 5 . Recompute the bipartite graph (call it H now) taking this into account, and then nd a minimum cover and a maximum matching in H , starting with the matching M . Does H have a perfect matching? 2. 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. 3. Give an example of a 3regular graph that does not have a pefect matching. (Note that such a graph cannot be bipartite.) 4. 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....
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 Spring '09
 M.PEI
 Math

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