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).
翻译:$L$ 应该是 $n_times n n$ 数列, 其左上方$r_time r$ subarrary 用不同的符号填充 $k$, 每一行最多一次, 每一列最多一次。 我们设置了必要和足够的条件, 以确保其余的 $L$ 的单元格可以填充, 这样每个符号在每一行最多一次, 每一列最多一次, 美元是对应主要对角的, 每个符号都以美元计数 。 规定每个符号的出现次数以美元计数 。 由 Cruse ( J. Complus. Theory Ser. A 16 (1974), 18- 22) 解决了每个符号发生次数为美元的情况。 左上方次为 $r\timets $, 不需要对称的情况, 由 Goldwassers et al. ( J. 组合. Ther.) (J.