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

TL;DR

This paper proves a lower bound on the number of filled cells in maximal orthogonal partial Latin squares and characterizes the structure achieving this bound for large orders.

Contribution

It establishes the minimum number of filled cells in such squares as at least one-third of the total, resolving a previous conjecture and describing the unique structure for large n.

Findings

F ≥ n^2/3 for the number of filled cells in maximal orthogonal partial Latin squares

For n ≥ 21, the minimum is exactly ⌈n^2/3⌉ and the structure is unique up to permutations

The result confirms the conjecture and characterizes the extremal configurations

Abstract

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.