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As the title states, we are given a bipartite undirected graph $G=(X\cup Y,E)$ such that every vertex $v\in V$ satisfies $d(v)=k$ for a constant $k$.

The general goal of the proof is to show that under these terms, the graph has a perfect matching (a subset of edges where all edges are disjoint in vertices, and the edges in said subset are reaching every vertex $v\in V$)

I wanted to use the proof of hall's marriage theorem, yet this requires that not only $|X|\leq |Y|$, but also that for every $S\subseteq X$, $|S|\leq |n(S)|$ where $n(S)$ is the group countaining neighbors of all vertices in $S$, which is the part I'm having a hard time proving.

I'm not sure how to explain that for every subset we choose $S\subseteq X$, the minimal size of $n(S)$ has to be $|S|$, which is due to the fact that every $v\in S$ has $k$ neighbors in $G$.

I've read about Konig's Theorem but I wasnt able to make the connection.

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Suppose $S\subseteq X$. Then $n(S)\subseteq Y$.

Consider all edges with one endpoint in $S$.

Since there are $k$ edges that start with any particular point in $S$ and all these edges are different for points in $S$, the number of those edges is $k|S|$.

Similarly, the number of all edges with one endpoint in $n(S)$ is $k|n(S)|$.

Since each edge with one endpoint in $S$ is also an edge with one endpoint in $n(S)$, we have $$ k|S|\le k|n(S)|,$$ which means $$ |S| \le |n(S)|.$$

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  • $\begingroup$ Thank you very much! $\endgroup$
    – Aishgadol
    Jan 19, 2023 at 9:16
  • $\begingroup$ You are welcome! $\endgroup$
    – John L.
    Jan 19, 2023 at 9:26

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