Balanced diagonals in frequency squares

2018/02/04 by Cavenagh, Nicholas, Mammoliti, Adam
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1802.01217

Abstract

We say that a diagonal in an array is \em λ-balanced if each entry occurs λ times. Let L be a frequency square of type F(n;λ^m); that is, an n× n array in which each entry from \1,2,… ,m\ 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.

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