The Intersections of Commutative Latin Squares

A Latin square of order \(n\) is an \(n \times n\) array such that each of the integers \(1, 2, \ldots, n\) (or any set of \(n\) distinct symbols) occurs exactly once in each row and each column. A Latin square \(L = [l_{i,j}]\) is said to be \underline{commutative} provided that \(l_{i,j} = l_{j,i}\) for all \(i\) and \(j\). Two Latin squares, \(L = [l_{i,j}]\) and \(M = [m_{i,j}]\), are said to have \underline{intersection} \(k\) if there are exactly \(k\) cells \((i,j)\) such that \(l_{i,j} = m_{i,j}\).

Let \(I[n] = \{0, 1, 2, \ldots, n^2-9, n^2-8, n^2-7, n^2-6,n^2\}\), \(H[n] = I[n] \cup \{n^2-7, n^2-4\}\), and \(J[n]\) be the set of all integers \(k\) such that there exists a pair of commutative Latin squares of order \(n$ which have intersection \(k\). In this paper, we prove that \(J[n] = I[n]\) for each odd \(n \geq 7\), \(J[n] = H[n]\) for each even \(n \geq 6\), and give a list of \(J[n]\) for \(n \leq 5\). This totally solves the intersection problem of two commutative Latin squares.