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# matching - Graph Theory Matchings and Hall’s Theorem COS...

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Unformatted text preview: Graph Theory: Matchings and Hall’s Theorem COS 341 Fall 2004 Definition 1 A matching M in a graph G ( V, E ) is a subset of the edge set E such that no two edges in M are incident on the same vertex, i.e. if { w, x } , { y, z } ∈ M , then the vertices w, x, y, z are distinct. The size of a matching M is the number of edges in M . For a graph G ( V, E ), a matching of maximum size is called a maximum matching. Definition 2 If M is a matching in a graph G , a vertex v is said to be M-saturated if there is an edge in M incident on v . Vertex v is said to be M-unsaturated if there is no edge in M incident on v . If G ( V 1 , V 2 , E ) is a bipartite graph than a matching M of G that saturates all the vertices in V 1 is called a complete matching (also called a perfect matching). When does a bipartite graph have a complete matching ? Given a graph, if we wanted to prove that the graph has a complete matching, we can simply give the edges in the matching....
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matching - Graph Theory Matchings and Hall’s Theorem COS...

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