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Lecture5 - CME 305 Discrete Mathematics and Algorithms...

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CME 305: Discrete Mathematics and Algorithms Instructor: Professor Amin Saberi ([email protected]) January 19 and 21, 2010 Lecture 5: Matching in Bipartite Graphs Definition: A graph G ( V, E ) is bipartite if we can partition V into two sets A and B such that e = ( i, j ) E , { i, j } * A and { i, j } * B . In other words, the two end points of an edge do not belong to the same set. We usually represent a bipartite graph by G ( A, B, E ). Definition: A matching M is a set of edges such that every vertex is incident to at most one edge in M . In other words, it is a set of “independent” edges that share no endpoints in common; M is a perfect matching iff | A | = | B | = | M | ; M is a maximal matching if it is not a subsent of any other matching. In other words, e E \ M , M ∪ { e } is not a matching; M is a maximum matching if there are no possible matching of a larger size. Question: Under what conditions bipartite graph G ( A, B, E ) has a perfect matching? Theorem 1 Hall’s Marriage Theorem (1935) Let G ( A, B, E ) be a bipartite graph such that | A | = | B | = n . G has a perfect matching iff S A, | S | ≤ | N ( S ) | where N ( S ) is the neighborhood of S , i.e. N ( S ) = { v B | ∃ u S : ( u, v ) E } . Proof: ”: It’s easy to see that if S A such that | S | > | N ( S ) | , G cannot have a perfect matching. ”: We prove this by strong induction on the size of A . The case n = 1 is trivial. We split the induction step into two cases: Case 1: S A, | S | < | N ( S ) | Pick an arbitrary vertex u A and match it to one of its neighbors v . Let A 1 = A \ { u } , B 1 = B \ { v } , and G 1 be the bipartite graph induced by A 1 and B 1 . S A 1 , | N G 1 ( S ) | ≥ | N G ( S ) | - 1 ≥ | S | , thus by assumption G 1 has a perfect matching M 1 . It is easy to see that M 1 ∪ { ( u, v ) } is a perfect matching of G .
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