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I need help with this question.

Let G be a bipartite graph with bipartition $(A,B)$, such that $|A| = |B| = 10$, and every vertex of G has degree at least 5. Show that G has a perfect matching.

We are given the following hint : *Show that if X is a vertex cover of G then either $|X \cap A| \geq 5$ and $|X \cap B| \geq 5$, or $A \subseteq X$, or $B \subseteq X.$

This somehow relates to König's Theorem which states that in a bipartite graph G, the minimum size of a vertex cover in G is equal to the maximum size of a matching in G.

Thanks.

1 Answers 1

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the hint clearly implies that $X$ must have $10$ vertices, which by König's theorem implies there is a matching with $10$ vertices.

So we just need to prove the hint. So suppose $X$ is a vertex cover that does not contain all of the vertices of $A$, we shall show that it contains at least $5$ vertices of $B$, this is obvious because any uncovered vertex in $A$ must have $5$ vertices in $B$, and all of these must belong to $X$. An analogous argument shows that if $X$ does not contain every vertex in $B$ it must contain at least $5$ vertices in $A$.

Conclusion: If $X$ does not contain $A$ and it does not contain $B$ it must contain at least $5$ vertices in each side.

We can also prove this using Hall's marriage theorem. Suppose that there is a subset $Y\subseteq A$ that has less than $|Y|$ neighbours. it follows that $|Y|$ is larger than $5$, but then every vertex in $B$ must have at least one vertex in $Y$, a contradiction, therefore halls condition holds and a matching that saturates $A$ exists.