Coloring in Combinatorial Design Generalizing Latin Square

Question

I have a question about a combinatorial design very similar to a Latin Square, which is arising out of an open problem in graph theory. The design is an $n \times n$ matrix whose entries we want to assign colors from $\lbrace 1, 2, \dots, n \rbrace$. A (symmetric) Latin Square is subject to the following constraints:

1) The matrix should be symmetric.

2) No row or column contains the same color more than once.

In my problem, we will want a symmetric $n \times n$ matrix, but we will allow some of the matrix entries to be identified into symmetric blocks. In other words, suppose there is an equivalence relation $\mathcal{A}$ on the coordinates $(a_{i,j})$ which satisfies the property that if $A \in \mathcal{A}$ and $a_{i,j} \in A$, then $a_{j,i} \in A$. The problem I am curious about is, if given an $n$ and an arbitrary such $\mathcal{A}$, is there an $n \times n$ matrix satisfying the following constraints:

1) The matrix is symmetric.

2) If $a, b \in A$ for some $A \in \mathcal{A}$, then the entries are assigned the same color, and

3) If two coordinates $a, b$ are in the same row or column as one another and they are the same color, then there is an $A \in \mathcal{A}$ such that $a, b \in A$.

In other words, it's a symmetric Latin Square 'up to equivalence by $\mathcal{A}$.'

Have these objects been studied before (with or without the symmetry condition), or are there any equivalent formulations? I expect that it is possible to construct such a design. I can only prove it in very special cases, though (e.g. where each row or column only 'intersects' at most one non-trivial equivalence class).

Answer 1

Hope I didn't misunderstand the question, but isn't this a counterexample (different letters correspond to different equivalence classes):

AAB
ACD
BDC   

Each pair of classes have representatives in a common row/column, hence at least four colours are needed.

Answer 2

If I understand correctly, these are symmetric frequency squares. Quoting from a random reference on the topic:

An $F(n;\lambda_1,\ldots,\lambda_m)$ frequency square is an $n \times n$ array consisting of the numbers $1,2,\ldots,m$ with the property that for each $i = 1,\ldots,m$, the number $i$ occurs exactly $\lambda_i$ times in each row and in each column.
_{Dénes and Mullen, Enumeration formulas for latin and frequency squares, Disc. Math., 1993}

Some Latin squares have the property that they're symbol-isotopic to their transpose. I.e., for some Latin squares $L$, you can permute the symbols of $L^T$ to obtain $L$. We can identify symbols to obtain the desired frequency squares.

To my knowledge, the only paper that talks about such Latin squares is: Mendis and Wanless, Autoparatopisms of Quasigroups and Latin Squares, J. Combin. Des., 2016. (There may be others that I'm unaware of.) Theorem 4.7 in this paper gives necessary and sufficient conditions for the existence of these Latin squares.