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# 数学代写|离散数学代写Discrete Mathematics代考|MATH215 The Main Theorem

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## 数学代写|离散数学代写Discrete Mathematics代考|The Main Theorem

Now we state and prove a fundamental theorem about perfect matchings. This will complete the solution of the problem about tribes and tortoises, and (with some additional work) of the problem about dancing at the prom (and some problems further down the road from the prom, as its name shows).

Theorem 10.3.1 (The Marriage Theorem) A bipartite graph has a perfect matching if and only if $|A|=|B|$ and for any subset of (say) $k$ nodes of $A$ there are at least $k$ nodes of $B$ that are connected to at least one of them.

This important theorem has many variations; some of these occur in the exercises. These were discovered by the German mathematician G. Frobenius, by the Hungarian D. König, the American P. Hall, and others.

Before proving this theorem, let us discuss one more question. If we interchange “left” and “right,” perfect matchings remain perfect matchings. But what happens to the condition stated in the theorem? It is easy to see that it remains valid (as it should). To see this, we have to argue that if we pick any set $S$ of $k$ nodes in $B$, then they are connected to at least $k$ nodes in $A$. Let $n=|A|=|B|$ and let us color the nodes in $A$ connected to nodes in $S$ black, the other nodes white (Figure 10.4). Then the white nodes are connected to at most $n-k$ nodes (since they are not connected to any node in $S$ ). Since the condition holds “from left to right,” the number of white nodes is at most $n-k$. But then the number of black nodes is at least $k$, which proves that the condition also holds “from right to left.”

Proof. Now we can turn to the proof of Theorem 10.3.1. We shall have to refer to the condition given in the theorem so often that it will be convenient to call graphs satisfying this conditions “good” (just for the duration of this proof). Thus a bipartite graph is “good” if it has the same number of nodes left and right, and any $k$ “left” nodes are connected to at least $k$ “right” nodes.

It is obvious that every graph with a perfect matching is “good,” so what we need to prove is the converse: Every “good” graph contains a perfect matching. For a graph on just two nodes, being “good” means that these two nodes are connected. Thus for a graph to have a perfect matching means that it can be partitioned into “good” graphs with 2 nodes. (To partition a graph means that we divide the nodes into classes, and keep an edge between two nodes only if they are in the same class.)

## 数学代写|离散数学代写Discrete Mathematics代考|How to Find a Perfect Matching

We have a condition for the existence of a perfect matching in a graph that is necessary and sufficient. Does this condition settle this issue once and for all? To be more precise: Suppose that somebody gives us a bipartite graph; what is a good way to decide whether it contains a perfect matching? And how do we find a perfect matching if there is one?

We may assume that $|A|=|B|$ (where, as before, $A$ is the set of nodes on the left and $B$ is the set of nodes on the right). This is easy to check, and if it fails, then it is obvious that no perfect matching exists, and we have nothing else to do.

One thing we can try is to look at all subsets of the edges, and see whether any of these is a perfect matching. It is easy enough to do so; but there are terribly many subsets to check! Say, in our introductory example, we have 300 nodes, so $|A|=|B|=150$; every node has degree 50 , so the number of edges is $150 \cdot 50=7500$; the number of subsets of a set of this size is $2^{7500}>10^{2257}$, a number that is more than astronomical.

We can do a little bit better if instead of checking all subsets of the edges, we look at all possible ways to pair up elements of $A$ with elements of $B$, and check whether any of these pairings matches only nodes that are connected to each other by an edge. Now the number of ways to pair up the nodes is “only” $150 ! \approx 10^{263}$. Still hopeless.

Can we use Theorem 10.3.1? To check that the necessary and sufficient condition for the existence of a perfect matching is satisfied, we have to look at every subset $S$ of $A$, and see whether the number of it neighbors in $B$ is at least as large as $S$ itself. Since the set $A$ has $2^{150} \approx 10^{45}$ subsets, this takes a much smaller number of cases to check than any of the previous possibilities, but still astronomical!

## 数学代写|离散数学代写Discrete Mathematics代考|The Main Theorem

(仅在本 证明期间）很方便。因此，如果二分图左右节点数相同，并且任意 $k^\mu$ 左”节点至少连接到 $k^{\prime}$ 正确”的节点。

## MATLAB代写

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