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h4a - MATH 4171 Graph Theory Homework Set IV-Solutions...

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MATH 4171 Graph Theory SPRING 2006 Homework Set IV -Solutions 1. Let G = ( X, Y, E ) be an r -regular ( r > 0) bipartite graph. Prove that G has a perfect matching. Proof. Let V = X Y . We prove that β 1 | V | 2 , which means that G has a matching M with | M | ≥ | V | 2 . Since M covers 2 | M | ≥ | V | vertices, it means that every vertex of G is incident with an edge in M , thus, by definition, M is a perfect matching. By K¨onig theorem, α = β 1 . So it is enough for us to show that α | V | 2 . By definition, we need to show that α = min {| S | : S V is a vertex cover of G } ≥ | V | 2 . For this, we only need to show that | S | ≥ | V | 2 , for every vertex cover S of G . Let S V be a vertex cover. Then V S is an independent set. It follows that E ( S, V S ), the set of edges between S and V S , consists of all edges that are incident with at least one vertex in V S . Since G is r -regular, this observation implies that | E ( S, V S ) | = r | V S | . On the other hand, the r -regularity also implies that each vertex in S is incident with at most r edges in E (
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