Balanced diagonals in frequency squares

Abstract

We say that a diagonal in an array is λ-balanced if each entry occurs λ times. Let L be a frequency square of type F(n;λ); that is, an n×n array in which each entry from {1,2,…,m=n∕λ} occurs λ times per row and λ times per column. We show that if m⩽3, L contains a λ-balanced diagonal, with only one exception up to equivalence when m=2. We give partial results for m⩾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.