Balanced diagonals in frequency squares |
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Authors: | Nicholas J Cavenagh Adam Mammoliti |
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Institution: | 1. Department of Mathematics, The University of Waikato, Private Bag 3105, Hamilton 3240, New Zealand;2. School of Mathematics and Statistics, UNSW Sydney, NSW 2052, Australia |
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Abstract: | We say that a diagonal in an array is -balanced if each entry occurs times. Let be a frequency square of type ; that is, an array in which each entry from occurs times per row and times per column. We show that if , contains a -balanced diagonal, with only one exception up to equivalence when . We give partial results for
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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Keywords: | 05B15 05C15 Frequency square Latin square Ryser’s conjecture Transversal |
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