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# hbs10 - CS 473 Algorithms Fall 2010 HBS 10 Problem...

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Unformatted text preview: CS 473: Algorithms, Fall 2010 HBS 10 Problem 1 . [k-Regular Bipartite Graphs] A k-Regular graph is an undirected graph where every vertex has degree k . We will prove that if a bipartite graph is k-Regular, then it has a perfect matching. First, recall the following definitions: Bipartite Graph : a graph whose vertices are partitioned into two independent sets, L and R . Matching : A matching in a graph G is a set of edges such that no two edges share a common vertex. Neighbors : Let v be a vertex. The neighbors of v , denoted by N ( v ) are the set of vertices connected to v . Hall’s Theorem : Let G = ( L ∪ R,E ) be a bipartite graph where | L | = | R | . Then G has a perfect matching if and only if for every subset X ⊆ L , | N ( X ) | ≥ | X | . For the following problems, let G = ( L ∪ R,E ) be a k-regular bipartite graph where | L | = | R | . 1. Show that the G has a perfect matching via Hall’s theorem....
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