MT4514: Graph Theory

Chapter 5Marriage, Matchings and Connectivity1. Hall’s Marriage TheoremMany problems involve matching objects from one set with objects in another.Problem 1.1. [Hall’s Marriage Problem] Given a set B of boys and a set G of girls,some of whom know each other, under what conditions can we marry off the boys sothat each boy marries a girl he knows? (We assume that “knowing” is symmetric,so that if a boy knows a girl then the girl knows him too!)We can represent this problem with a bipartite graph with vertex sets B and G,where we put an edge joining a boy and a girl if they know each other.Example 1.2. In this graph a matching is possible.In this graph a matching is not possible, as boys 1, 3 and 5 only know 2 girls betweenthem.Definition 1.3. Two edges of a (not necessarily bipartite) graph are called independentif they do not share a vertex. A perfect matching of a bipartite graph Gis a set of independent edges such that every vertex is incident with one of them.Let G be bipartite, with parts V 1 and V 2 . A matching from V 1 to V 2 is a set ofindependent edges incident to every vertex in V 1 .Thus Hall’s Marriage Problem asks if there is a perfect matching from the girlsto the boys. Clearly, if there is some set of k boys that know fewer than k girls,then no complete matching is possible. Hall’s Marriage Theorem says (surprisingly)that this is the only situation where a matching is not possible.Theorem 1.4 (Hall’s Marriage Theorem) 1. There is a solution to Hall’sMarriage Problem 1.1 if and only if every set of k boys knows at least k girls,for 1 ≤ k ≤ |B|.33