Bipartite graph with big degrees

Let \( G \) be a bipartite graph with \( 2n \) vertices such that each of its classes of bipartition has \( n \) vertices.

Variant
Assume that the minimum degree in \( G \) is at least \( n / 2 \). Prove that \( G \) contains a perfect matching.
Hint
Use Hall’s theorem. To verify the Hall condition for the neighborhood of \( k \) elements, distinguish two cases according to whether \( k \leq n / 2 \) or not.
Solution
Let’s denote by \( A, B \) the classes of bipartition of the bipartite graph.

If a set \( I \subset A \) contains at most \( n / 2 \) vertices, then each has a degree of at least \( n / 2 \) and the Hall condition holds.

On the other hand, if \( I> \frac{n}{2} \), then \( N (I) = B \), since each of the vertices of \( B \) must have at least one neighbor in \( I \), because \( |A \setminus I | <\frac{n}2 \).

The Hall condition holds, and therefore there is a perfect matching.
Variant
Must \( G \) contain a perfect matching if the minimum degree is \(\lceil n / 2 \rceil-1 \)?
Hint
Seek for a (disconnected) counterexample.
Answer
In this case, \( G \) may not contain a perfect matching, for example, a disjoint union of two \( K_{1{,}2} \). Here \( n = 3 \), and the minimum degree is \( 1 = \lceil3 / 2 \rceil-1 \).