# 4204hw2 - S in the graph G to be a vertex cover if all...

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Combinatorics II (MAD 4204) — Spring 2011 Due Wednesday 1/26/2011 Homework #2 1 Let G be a bipartite graph, and suppose that vw is an edge of G . Prove either v or w has the property that every maximum matching of G contains an edge incident to this vertex. Note: Of course, every maximum matching of G contains an edge incident to either v or w . This question is asking for something stronger, however. In Homework #1, we deFned a set of vertices
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Unformatted text preview: S in the graph G to be a vertex cover if all edges of G have at least one of their vertices in S . Let τ ( G ) denote the size of the smallest vertex cover of G and let ν ( G ) denote the size of its maximum matching. We proved the following theorem: K¨ onig’s Theorem. In any bipartite graph G , ν ( G ) = τ ( G ). 2 Prove Hall’s Marriage Theorem from K¨ onig’s Theorem....
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## This note was uploaded on 12/10/2011 for the course MAD 4204 taught by Professor Vetter during the Spring '11 term at University of Florida.

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