Unformatted text preview: t 1 or t 5 . Recompute the bipartite graph (call it H now) taking this into account, and then ﬁnd 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 3-regular 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
- Math, Bipartite graph, perfect matching, Dulmage–Mendelsohn decomposition, G. Hence