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T9 - G that avoids e Problem 3 Let G be a bipartite graph...

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Problem 1: Let M be a maximum matching of a graph G . Let N be any other matching of G . Let H be the subgraph of G whose edges are ( M N ) \ ( M N ) and whose vertices are the vertices of G . Show that if every component of H is a cycle then N is also a maximum matching of G . Problem 2: Let G be a graph, and e be a bridge of G . Show that if there is one perfect matching of G that contains e , then every perfect matching of G contains e . Give an example where there is a maximum matching of G that contains e , but also a maximum matching of
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Unformatted text preview: G that avoids e . Problem 3: Let G be a bipartite graph with q edges and maximum degree Δ. Prove that G has a matching of size at least q Δ . Give an example of a graph G (not bipartite) that has q edges, maximum degree Δ, and no matching of size at least q Δ . Problem 4: Exercise 6 from Problem Set 7.2 in the course notes. (Or some other example of the bipartite maximum matching algorithm if your instructor did that one in class.) 1...
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