On the minimum number of entries in a pair of maximal orthogonal partial Latin squares

It is shown that if $F$ denotes the number of filled cells in a superimposed pair of maximal orthogonal partial Latin squares of order $n$, then $F\ge n^2/3$. This resolves a conjecture raised in an earlier paper by the current authors. It is also shown that, for $n\ge 21$, the least possible number of filled cells in a pair of maximal orthogonal partial Latin squares is $\lceil n^2/3 \rceil$, and that the structure that achieves this bound is unique up to permutations of rows, columns and entries.

💡 Research Summary

The paper investigates the smallest possible number of filled cells in a pair of maximal orthogonal partial Latin squares (MOPLS) of order n. By interpreting Latin squares as transversal designs and as decompositions of complete multipartite graphs, the authors translate the problem into a graph‑theoretic one: an OPLS corresponds to a collection of K₄ copies that partition a subgraph of the complete 4‑partite graph Kₙ,ₙ,ₙ,ₙ, and a maximal OPLS is exactly a configuration whose complement is K₄‑free. Using known extremal results for K₄‑free multipartite graphs (Jin, Pfender, and others), they show that if the density between any two parts exceeds 2/3 then a K₄ must appear in the complement, contradicting maximality. Since the density is (n²−F)/n², this yields the lower bound F ≥ n²/3 for the number of filled cells.

The authors then prove that the bound is tight. Lemma 2.2 analyses a d × d array with a maximal partial transversal of empty cells, showing that all cells outside the transversal must be filled and that the sum of row‑ and column‑frequencies satisfies f_r(i)+f_c(j) ≥ 2d−t. Applying this to a maximal OPLS forces every row, column, first symbol and second symbol to appear exactly n/3 times when F = n²/3, and the array can be partitioned into four blocks: two full Latin squares of orders m and n−m (with m = n/3) and two empty blocks. This structure is forced by the equality case of the bound.

For constructive optimality, the paper presents an explicit family of MOPLS for all n ≥ 21. Write n = 3s + r with r ∈ {0,1,2}. Start with an empty n × n grid and place three mutually orthogonal Latin squares along the main diagonal: a top‑left s × s square using symbols 1…s, a central square of size s (or s+1) using symbols s+1…2s (or 2s+1), and a bottom‑right square using the remaining symbols. The three squares are orthogonal because they occupy disjoint symbol sets and distinct diagonal blocks. The total number of filled cells is

• 3s² = n²/3 when r = 0,
• 2s² + (s+1)² = (n² + 2)/3 when r = 1,
• s² + 2(s+1)² = (n² + 2)/3 when r = 2, which equals ⌈n²/3⌉ for all n ≥ 21 (since s ≥ 7). A local maximality argument shows that any empty cell lies in a row and column where the possible first and second symbols are already paired elsewhere, so no additional entry pair can be inserted.

The paper also notes the trivial exceptions n = 2 and n = 6, where a full pair of orthogonal Latin squares exists, giving F = n². Apart from these, the lower bound is exact and the construction described is the unique (up to permutations of rows, columns, and symbols) optimal configuration.

In summary, the authors combine combinatorial design theory with extremal graph theory to settle a conjecture: any maximal orthogonal partial Latin square pair must contain at least n²/3 filled cells, and for n ≥ 21 this bound is attained uniquely by a diagonal block construction. This result closes a gap in the theory of partial Latin squares, provides a clear structural characterisation of extremal examples, and links the problem to well‑studied density conditions in multipartite graphs.