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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.
Type
Journal Article
Type of thesis
Series
Citation
Cavenagh, N. J., & Mammoliti, A. (2018). Balanced diagonals in frequency squares. Discrete Mathematics, 341(8), 2293–2301. https://doi.org/10.1016/j.disc.2018.04.029
Date
2018
Publisher
Elsevier
Degree
Supervisors
Rights
This is an author’s accepted version of an article published in the journal: Discrete Mathematics. © 2018 Elsevier.