Cavenagh, Nicholas J.Mammoliti, Adam2018-09-032018-08-012018-09-032018Cavenagh, N. J., & Mammoliti, A. (2018). Balanced diagonals in frequency squares. Discrete Mathematics, 341(8), 2293–2301. https://doi.org/10.1016/j.disc.2018.04.0290012-365Xhttps://hdl.handle.net/10289/12055We 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.application/pdfenThis is an author’s accepted version of an article published in the journal: Discrete Mathematics. © 2018 Elsevier.Science & TechnologyPhysical SciencesMathematicsFrequency squareLatin squareRyser's conjectureTransversalLATIN SQUARESORTHOGONAL MATESJournal Article10.1016/j.disc.2018.04.0291872-681X