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Question: Suppose we have a $n \times n$ grid that has $k$ black cells in each row and $k$ black cells in each column $(1\le k \le n)$. Prove this grid has a scattered black diagonal.

This question was given to me for graph theory but I can't link it to that subject. I tried to solve it step by step:
for $1 \times 1$ and $2 \times 2$ it's obvious. for a $3 \times 3$ grid, our $k$ could be $1, 2$ or $3$. Visually, the existence of the diagonal is obvious to me, but I have no idea how I can prove this mathematically.
I would appreciate it if someone could prove this.

ArithEgo
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    How do you define exactly "scattered diagonal"? It may be confusing for non-native english speaker. – Alejandro Bergasa Alonso Dec 27 '21 at 18:03
  • Assuming "scattered diagonal" means "set of $n$ squares with no two in the same row or column", then this question is equivalent to asking whether a $k$-regular bipartite graph always has a perfect matching. This has been asked and answered many times on MSE, e.g: https://math.stackexchange.com/questions/1805181/prove-that-a-k-regular-bipartite-graph-has-a-perfect-matching. – Mike Earnest Dec 27 '21 at 18:06
  • @MikeEarnest I have no further explanation on this question and I was just given this but I assumed this means: if we move the black squares in their row or in their column, we surely can make a diagonal for our square. (because if it was as you explained then the $1 \lt k \lt n$ would be nonsense) – ArithEgo Dec 27 '21 at 18:30
  • I cannot understand what you mean when you say "if we move the black squares in their row or column, we surely can make a diagonal". Can you give an example? When $n=3$, and the squares are numbered like a telephone keypad, what would be the number of the broken diagonals? – Mike Earnest Dec 27 '21 at 18:34
  • @MikeEarnest $124689$ for $k = 2$ and $249$ for $k=1$ – ArithEgo Dec 27 '21 at 18:40
  • @MikeEarnest how do we say that this problem is like the one you recalled? – ArithEgo Dec 27 '21 at 20:25
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    Given your grid, we make a bipartite graph as follows. There is one vertex for each row, and one vertex for each column. A row vertex is connected to a column vertex by an edge if the cell at that row and column is black. Since each row and column has $k$ black squares, the bipartite graph is $k$-regular. A perfect matching is equivalent to a scattered diagonal. – Mike Earnest Dec 27 '21 at 21:10

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