NOTES ON MATCHING
Notice that the marriage theorem is a subcase of Proposition 1.8, where N =
|L| = |R| and DL (G) = 0. Both theorems say that in this case, the matching
number is N , or in other words there exists a complete matching.
It is easy to see that the marriage condition is necessary for a complete matching,
but Hall’s marriage theorem asserts that it is also sufficient.
Proof of Hall’s Marriage Theorem. Since necessity is easy to see, we need to prove
that the marriage condition is also sufficient. That is to say, if the marriage condition holds, then there exists a complete matching.
We will use induction to prove our desired result. Given a balanced bipartite
graph G, assume that the marriage condition holds. We will induct on r, the size
of a subset S of left vertices of our graph G. What we need to do is show that if
the marriage theorem holds for r, then it also holds for r + 1, and eventually the
entire graph G. But first the base case, where r = 1.
If r = 1 and the marriage condition holds true for any subset of size one, then
clearly it can be paired with a right vertex. Then we have a complete matching for
any one vertex. Base case is done.
Now for the inductive step. Assume that for any r-sized subset S, the marriage
condition holds and so does the marriage theorem. Now we need to show, based on
the previous assumptions, that the marriage theorem also holds true for r + 1. Let
the set of vertices that S connects to be denoted as S 0 . Now consider any (r + 1)th
left vertex, vr+1 . Now we split the problem into three cases.
Case 1. The vertex vr+1 is connected to some vertex not in S 0 . Then clearly
there exists a complete matching among these (r + 1) vertices. This is illustrated
in Figure 4.
Figure 4. Case 1
Case 2. The vertex vr+1 is only connected to vertices in S 0 , and |S 0 | = |S|. In
this case, it is clear that this violates the marriage condition which we have assumed
to be true for our entire graph G. So this case is impossible.
Case 3. The vertex vr+1 is only connected to vertices in S 0 , and |S 0 | > |S|.
Then we use our assumption that for every r-sized subset we can find a complete
matching. In particular, we look at every r-sized subset that includes (r − 1)
members of S plus vertex vr+1 . Now what we must show is that given this property
(that all these r-sized subsets have a complete matching), this implies that we can