Balanced diagonals in frequency squares

作者：Cavenagh Nicholas J^*; Mammoliti Adam

来源：Discrete Mathematics, 2018, 341(8): 2293-2301.

DOI：10.1016/j.disc.2018.04.029

• Frequency square
• Latin square
• Ryser's conjecture
• Transversal

We say that a diagonal in an array is lambda-balanced if each entry occurs lambda times. Let L be a frequency square of type F(n; lambda); that is, an n x n array in which each entry from {1, 2,..., m = n/lambda} occurs lambda times per row and lambda times per column. We show that if m 3, L contains a lambda-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.

• 出版日期2018-8