Regular bipartite graphs theorem 3 every r regular

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Regular bipartite graphs Theorem 3. Every r -regular bipartite graph ( r 1) has a perfect match- ing. Proof. Let G be an r -regular bipartite graph with bipartition { V 1 , V 2 } . Then the size of G is equal to both r | V 1 | and r | V 2 | . Hence | V 1 | = | V 2 | . Thus, it suffices to prove that G has a matching under which all vertices of V 1 are matched.
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Let ∅ 6 = S V 1 . Since G is r -regular, the number of edges joining the vertices of S and the vertices of N ( S ) is equal to r | S | . All these edges are incident with the vertices of N ( S ). However, the total number of edges incident with the vertices of N ( S ) is equal to r | N ( S ) | . Hence r | N ( S ) | ≥ r | S | . Therefore, | N ( S ) | ≥ | S | and by K¨ onig-Hall Theorem, G contains a matching that matches all vertices in V 1 . Systems of distinct representatives Definition 4. A collection A 1 , A 2 , . . . , A n , n 1, of finite nonempty sets (not necessarily distinct) is said to have a a system of distinct representatives (SDR) if there exists a set { a 1 , a 2 , . . . , a n } of distinct elements such that a i A i for 1 i n . Theorem 4 (Hall’s Theorem) . A collection { A 1 , A 2 , . . . , A n } of finite nonempty sets (not necessarily distinct) has a system of distinct representatives if and only if for any 1 k n the union of any k of these sets contains at least k elements. Proof. Set A = { A 1 , A 2 , . . . , A n } . Define the bipartite graph G with bipar- tition {A , n i =1 A i } such that A i is adjacent to a ∈ ∪ n i =1 A i if and only if a A i . Then A has an SDR if and only if G has a matching under which all vertices A 1 , A 2 , . . . , A n are matched, which, by K¨ onig-Hall Theorem, is true if and only if for any J ⊆ { 1 , 2 , . . . , n } , | ∪ i J A i | ≥ | J | . Remarks onig-Hall Theorem is equivalent to Hall’s Theorem. In the marriage problem, everyone can be married every set of k women are collectively compatible with at least k men every set of k men are collectively compatible with at least k women. Augmentation along an augmenting path Definition 5. If M is a matching in G and P is an augmenting path with respect to M , by swapping edges in and out of M in P we obtain a matching M 0 with | M 0 | = | M | + 1, namely M 0 = ( M - M E ( P )) ( E ( P ) - M E ( P )) . This operation is called augmentation (more specifically, augmentation of M along the augmenting path P ). All vertices on P are matched under M 0 , whilst the two end-vertices of P are unmatched under M . Most algorithms for finding maximum matchings are based on the following idea: Keep augmenting until there is no augmenting path. We will learn how to implement this systematically for bipartite graphs. Alternating trees Definition 6. Let M be a matching in G and u an unmatched vertex with respect to M . An alternating tree in G with respect to M and root u is a tree T such that all ( u, v )-paths in T are alternating with respect to M .
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