Decomposition of bi-colored square arrays into balanced diagonals

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.