Balanced diagonals in frequency squares

TL;DR

This paper investigates the existence of balanced diagonals in frequency squares, establishing results for small cases, partial results for larger cases, and proposing a generalization related to Ryser's conjecture on transversals in Latin squares.

Contribution

It proves the existence of balanced diagonals in frequency squares for small m, provides partial results for larger m, and introduces a new conjecture generalizing Ryser's conjecture.

Findings

Balanced diagonals exist for m ≤ 3 with one exception at m=2.

Partial results for m ≥ 4.

A proposed generalization of Ryser's conjecture.

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.