Balanced diagonals in frequency squares

Abstract

We say that a diagonal in an array is {\em $\lambda$-balanced} if each entry occurs $\lambda$ times. Let $L$ be a frequency square of type $F(n;\lambda^m)$; that is, an $n\times n$ array in which each entry from $\{1,2,\dots ,m\}$ occurs $\lambda$ times per row and $\lambda$ times per column. We show that if $m\leq 3$, $L$ contains a $\lambda$-balanced diagonal, with only one exception up to equivalence when $m=2$. We give partial results for $m\geq 4$ and suggest a generalization of Ryser's conjecture, that every latin square of odd order has a transversal. Our method relies on first identifying a small substructure with the frequency square that facilitates the task of locating a balanced diagonal in the entire array.