Decomposition of bi-colored square arrays into balanced diagonals

TL;DR

This paper establishes necessary and sufficient conditions for partitioning a bi-colored square array into diagonals with both colors, using Latin square completion results, and identifies minimum color counts for guaranteed partitions.

Contribution

It introduces a precise criterion for diagonally partitioning bi-colored arrays and links the problem to Latin square completion, providing new insights into array decomposition.

Findings

Partition exists if each color appears at least 2n-1 times.

Necessary and sufficient conditions for diagonal partitions.

Connection to Latin square completion results.

Abstract

Given an $n\times n$ array $M$ ($n\ge 7$), where each cell is colored in one of two colors, we give a necessary and sufficient condition for the existence of a partition of $M$ into $n$ diagonals, each containing at least one cell of each color. As a consequence, it follows that if each color appears in at least $2n-1$ cells, then such a partition exists. The proof uses results on completion of partial Latin squares.