论文标题
Cruse定理的另一个证明和完成部分拉丁正方形的新必要条件（第3部分）
Another proof of Cruse's theorem and a new necessary condition for completion of partial Latin squares (Part 3.)
论文作者
论文摘要
部分拉丁订单$ n $的部分拉丁广场可以由$ 3 $维的棋盘大小$ n \ times n \ times n $，最多最多$ n^2 $ n^2 $ nontakearing Rooks。基于此表示形式，我们采用统一方法来证明M. Hall，Ryser's和Cruse的定理完成部分拉丁正方形。在此证明的帮助下，我们将Cruse定理的范围扩展到紧凑的砖块，这些砖块似乎与它们的环境无关。 如果不丢失任何完成，您可以用rok替换DOT，如果DOT必须成为Rook，或者您可以消除已知不会成为菜鸟的点。因此，我们介绍了尽可能多次重复的主要和次要扩展程序。如果过程不确定是否可以完成PLSC，则可以针对所得PLSC的点结构（错误条件）制定新的必要条件。
A partial Latin square of order $n$ can be represented by a $3$-dimensional chess-board of size $n\times n\times n$ with at most $n^2$ non-attacking rooks. Based on this representation, we apply a uniform method to prove the M. Hall's, Ryser's and Cruse's theorems for completion of partial Latin squares. With the help of this proof, we extend the scope of Cruse's theorem to compact bricks, which appear to be independent of their environment. Without losing any completion you can replace a dot by a rook if the dot must become rook, or you can eliminate the dots that are known not to become rooks. Therefore, we introduce primary and secondary extension procedures that are repeated as many times as possible. If the procedures do not decide whether a PLSC can be completed or not, a new necessary condition for completion can be formulated for the dot structure of the resulting PLSC, the BUG condition.
