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Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section11.1: Parabolas

Problem 69E

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1. Let (G, X, Y) be a bipartite graph. Prove that there exist a matching M of and a subset U
of X such that
|M| ≥ |N(U)| + |X| − |U|.
(Hint: Use König's theorem)

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Let G = (V, E) be a finite bipartite graph with bipartition (A, B) where |A| = |B|. We say that G satisfies the marriage condition iff for every SCA, ING(S) ≥ |S| where NG(S) = {be B: (3a € S)({a, b} = E)} It should be clear that if G has a perfect matching, then it satisfies the marriage condition. Hall's theorem says that the converse is also true.
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