论文标题
Ryser定理的对称$ρ$ -LATIN Squares
Ryser's Theorem for Symmetric $ρ$-latin Squares
论文作者
论文摘要
令$ l $为$ n \ times n $阵列，其顶部剩下的$ r \ times r $ subaray填充了$ k $不同的符号，每个符号最多一次发生在每一行中，每列最多一次。我们建立了必要和充分的条件，以确保$ l $的其余单元可以填充，以使每个符号最多一次发生在每一行，每一列最多一次，每列中最多一次，$ l $相对于主对角线是对称的，并且每个符号在$ l $中发生了规定的次数。每个符号发生的规定次数为$ n $的情况是由Cruse（J。Combin。理论Ser。A16（1974），18-22）解决的，而Goldwasser等人则解决了左上列表为$ r \ times n $的情况。 （J. Combin。理论Ser。A130（2015），26-41）。我们的结果也可以指定主角的条目，从而导致安德森·霍夫曼定理的扩展（碟片学年。
Let $L$ be an $n\times n$ array whose top left $r\times r$ subarray is filled with $k$ different symbols, each occurring at most once in each row and at most once in each column. We establish necessary and sufficient conditions that ensure the remaining cells of $L$ can be filled such that each symbol occurs at most once in each row and at most once in each column, $L$ is symmetric with respect to the main diagonal, and each symbol occurs a prescribed number of times in $L$. The case where the prescribed number of times each symbol occurs is $n$ was solved by Cruse (J. Combin. Theory Ser. A 16 (1974), 18-22), and the case where the top left subarray is $r\times n$ and the symmetry is not required, was settled by Goldwasser et al. (J. Combin. Theory Ser. A 130 (2015), 26-41). Our result allows the entries of the main diagonal to be specified as well, which leads to an extension of the Andersen-Hoffman's Theorem (Annals of Disc. Math. 15 (1982) 9-26, European J. Combin. 4 (1983) 33-35).
