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Published online by Cambridge University Press: 05 November 2011
Properties of maximum matchings
A set of edges in a graph G is called a matching if no two edges have a common end vertex. A matching with the largest possible number of edges is called a maximum matching.
Many discrete problems can be formulated as problems about maximum matchings. Consider, for example, probably the most famous:
A set of boys each know several girls, is it possible for the boys each to marry a girl that he knows?
This situation has a natural representation as the bipartite graph with bipartition (V1, V2), where V1 is the set of boys, V2 the set of girls and an edge between a boy and a girl represents that they know one another. The marriage problem is then the problem: does a maximum matching of G have |V1| edges?
Let M be a matching of a graph G. A vertex v is said to be covered, or saturated by M, if some edge of M is incident with v. We shall also call an unsaturated vertex free. A path or cycle is alternating, relative to M, if its edges are alternately in E(G)\M and M. A path is an augmenting path if it is an alternating path with free origin and terminus. Throughout this, and the following section, we shall identify a path P or a cycle C with the set of edges it embodies, and write |P| and |C| for the cardinalities of these sets of edges.
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