Critical Sets of Full n-Latin Squares

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• 作者：Nicholas J. Cavenagh ; Vaipuna Raass
• 关键词：Full Latin square ; Latin square ; Defining set ; Critical set
• 刊名：Graphs and Combinatorics
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：32
• 期：2
• 页码：543-552
• 全文大小：375 KB
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• 作者单位：Nicholas J. Cavenagh (1)
  Vaipuna Raass (1)
  
  1. Department of Mathematics, The University of Waikato, Private Bag 3105, Hamilton, 3240, New Zealand
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  Engineering Design
  
• 出版者：Springer Japan
• ISSN：1435-5914

文摘

The full n-Latin square is the \(n\times n\) array with symbols \(1,2,\dots ,n\) in each cell. In a way that is analogous to critical sets of full designs, a critical set of the full n-Latin square can be used to find a defining set for any Latin square of order n. In this paper we study the size of the smallest critical set for a full n-Latin square, showing this to be somewhere between \((n^3-2n^2+2n)/2\) and \((n-1)^3+1\). In the case that each cell is either full or empty, we show the size of a critical set in the full n-Latin square is always equal to \(n^3-2n^2-n\).