To illustrate the ideas of perfect matching and Hall’s Marriage Theorem we start this section with the following two problems:
Problem 1
Suppose four women and four men want to find love so they join a dating agency. The agency compares applications and tries to find common interests. Two women, Jo and Mary, enjoy skiing, travelling, musical theatre and music. Kate likes basketball and Sue likes travelling, but hates musical theatre. Tom likes skiing and sport, John likes travelling and basketball, but hates other sports, and Steve and Ralph like reading, musical theatre, swimming and skiing. Is it possible to match the men and women into couples so that each couple share a common interest?
Sue Mary Kate Jo Women John Ralph Steve Tom Men
Figure 3.1: A bipartite graph describing the common interests between the four women and the four men. On the left, the set of vertices represents the women and on the right the set of vertices represents the men. Two vertices are joined by an edge if the people they represent share a common interest.
Stacy Casey Alex Candidates
Data entry person
Receptionist
Secretary Jobs
Figure 3.2: A bipartite graph describing the work experience of the candidates. The two sets of vertices represent the candidates and jobs. Two vertices are joined by an edge if the candidate has experience in that job.
Problem 2
A company needs to hire a secretary, data entry person and receptionist. There are three can- didates who have experience and qualifications for some or all of these jobs: Alex has worked as a receptionist, data entry person and secretary. Casey has experience in data entry and reception, and Stacy has experience as both a secretary and a receptionist. The company has one position for each job. Is it possible to match each candidate to a job they have experience with so all of them are hired?
We may model each of the situations above using a bipartite graph. For the first problem, the two sets of vertices represent the women and men respectively, and the edges represent a shared interest, see Figure 3.1. For the second problem, let the two sets of vertices represent the candidates and jobs respectively, and the edges represent experience in that job, see Figure 3.2. The problem is then to pair the women and men, and the candidates and jobs, so each pair is joined by an edge.
In order to solve these kinds of matching problems and find a perfect matching, Hall gave us a necessary and sufficient condition in a bipartite graph to determine whether a perfect
matching is possible. Note that Hall’s condition applies when the two sets have the same size. It is easy to see that the condition is necessary, so the main mathematical content of the theorem is that the condition is also sufficient. Hall’s Theorem is also commonly called the “Marriage Theorem” because it is frequently stated in terms of matching men and women, as we have done above.
Theorem 3.1.2(Hall 1935). LetGbe a bipartite graph with partsAandB such that|A|=|B|. The following are equivalent:
1. There is a perfect matching from A to B.
2. For all H ⊆A, we have |H| ≤ |N(H)| (Hall’s condition), where N(H)⊆B is the set of vertices in B adjacent to a vertex in H.
If the stronger condition |H|<|N(H)| holds for allH, then for any a ∈A and X ∈N(A)
there exists a perfect matching such that a is matched with X. More generally, if |H| <
|N(H)| −r for all H then we expect to be able to extend a partial matching at any r vertices of A to a matching in all ofA.
Hall’s Theorem was first proved by Philip Hall [5] in 1935. Hall’s condition is clearly nec- essary because, if there is a subset H of A for which |N(H)|<|H|, then there are not enough vertices of B available to match the vertices in H. Note that Hall’s Theorem does not tell us how to find a perfect matching.
Now, we illustrate Hall’s Theorem using the problems outlined above.
Problem 1
Consider the graph shown in Figure 3.1. To find out whether the agency is able to match all of the applicants into couples that share a common interest, we need to check that for every subset H of the women, the neighbourhood N(H) is at least as large as H. However, for
H ={Kate, Sue} ⊆ {Jo, Kate, Mary, Sue},
we have
|N(H)|=|{John}|= 1<2 =|{Kate, Sue}|=|H|.
Therefore, we can conclude that by Hall’s Theorem, the agency is not able to match the four women with the four men so that they share a common interest.
Problem 2
Consider the graph shown in Figure 3.2. We again need to check that for every subset H of the candidates, the neighbourhood N(H)is at least as large as H. For example, for
H ={Casey, Stacy} ⊆ {Alex, Casey, Stacy},
we have
|N(H)|=|{Secretary, Data entry person, Receptionist}|= 3>2 =|{Casey, Stacy}|=|H|.
By exhaustively checking the remaining six nonempty subsets H ⊆A, we find that in each case, the neighbourhood N(H) is at least as large as H. Therefore, by Hall’s Theorem, the company is able to hire all the candidates in a position they are experienced in. Figure 3.3 shows one possible match.
Stacy Casey Alex Candidates
Data entry person
Receptionist
Secretary Jobs
Figure 3.3: A bipartite graph describing one possible matching from candidates to jobs.