I am finding references for the following problem:

We call a $n\times n$ 0-1 matrix permutation if there are exactly one $1$ in each row/column. Suppose $A$ is a 0-1 matrix of size $n\times n$ in which there are exactly $n$ $1$s.

Conjecture. There exist permutations $P_i,Q_i$ such that $\sum_{i=1}^n P_iAQ_i = J$, where $J$ denotes the "all-$1$" matrix.

I am sure that we have a famous similar conjecture on non-bipartite graphs, i.e., $K_n$, in which we only consider the case that monochromatic parts are trees. But I just cannot recall its name...

I am finding references for the following problem:

We call a $n\times n$ 0-1 matrix permutation if there are exactly one $1$ in each row/column. Suppose $A$ is a 0-1 matrix of size $n\times n$ in which there are exactly $n$ $1$s.

Conjecture. There exist permutations $P_i,Q_i$ such that $\sum_{i=1}^n P_iAQ_i = J$, where $J$ denotes the "all-$1$" matrix.

I am sure that we have a famous similar conjecture on non-bipartite graphs, i.e., $K_n$, in which we only consider the case that monochromatic parts are trees. But I just cannot recall its name...

Update. (Mar 21, 2019) The name I forgot is Ringel's Conjecture.

I am finding references for the following problem:

We call a $n\times n$ 0-1 matrix permutation if there are exactly one $1$ in each row/column. Suppose $A$ is a 0-1 matrix of size $n\times n$ in which there are exactly $n$ $1$s.

Conjecture. There exist permutations $P_i,Q_i$ such that $\sum_{i=1}^n P_iAQ_i = J$, where $J$ denotes the ``all-$1$'' matrix.

I am sure that we have a famous similar conjecture on non-bipartite graphs, i.e., $K_n$, in which we only consider the case that monochromatic parts are trees. But I just cannot recall its name...

I am finding references for the following problem:

We call a $n\times n$ 0-1 matrix permutation if there are exactly one $1$ in each row/column. Suppose $A$ is a 0-1 matrix of size $n\times n$ in which there are exactly $n$ $1$s.

Conjecture. There exist permutations $P_i,Q_i$ such that $\sum_{i=1}^n P_iAQ_i = J$, where $J$ denotes the "all-$1$" matrix.

I am sure that we have a famous similar conjecture on non-bipartite graphs, i.e., $K_n$, in which we only consider the case that monochromatic parts are trees. But I just cannot recall its name...

I am finding references for the following problem:

We call a $n\times n$ 0-1 matrix permutation if there are exactly one $1$ in each row/column. Suppose $A$ is a 0-1 matrix of size $n\times n$ in which there are exactly $n$ $1$s.

Conjecture. There exist permutations $P_i,Q_i$ such that $\sum_{i=1}^n P_iAQ_i = J$ where $J$ denotes the ``all-$1$'' matrix.

I am sure that we have a famous similar conjecture on non-bipartite graphs, i.e., $K_n$, in which we only consider the case that monochromatic parts are trees. But I just cannot recall its name...

I am finding references for the following problem:

We call a $n\times n$ 0-1 matrix permutation if there are exactly one $1$ in each row/column. Suppose $A$ is a 0-1 matrix of size $n\times n$ in which there are exactly $n$ $1$s.

Conjecture. There exist permutations $P_i,Q_i$ such that $\sum_{i=1}^n P_iAQ_i = J$, where $J$ denotes the ``all-$1$'' matrix.

I am sure that we have a famous similar conjecture on non-bipartite graphs, i.e., $K_n$, in which we only consider the case that monochromatic parts are trees. But I just cannot recall its name...