发表机构
Institut für Mathematik, Freie Universität Berlin; School of Mathematics, University of Birmingham; School of Mathematics and Statistics, University of New South Wales; Institute of Science and Technology Austria (ISTA)（柏林自由大学数学研究所; 伯明翰大学数学学院; 新南威尔士大学数学与统计学院; 奥地利科学与技术研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文改进了部分拉丁方完备化的条件，证明当每行每列及每个符号使用次数不超过0.231n时可完备化，优于此前0.08n的界，采用放电策略与平衡性方法。
AI 中文摘要
一个$n$阶拉丁方是一个$n \ imes n$的阵列，填充了$n$个符号，使得每个符号在每一行和每一列中恰好出现一次。一个$n$阶部分拉丁方是一个$n \ imes n$的阵列，其单元格要么为空，要么以这样的方式填充：每个符号在每一行和每一列中至多出现一次，并且使用的不同符号至多$n$个。1983年，Daykin和Häggkvist猜想：每个部分拉丁方，如果其每一行和每一列包含至多$n/4$个符号，且每个符号被使用至多$n/4$次，则可以完备化为一个拉丁方。我们证明：每个部分拉丁方，如果其每一行和每一列包含至多$0.231n$个符号，且每个符号被使用至多$0.231n$次，则可以完备化为一个拉丁方，显著改进了先前由Fu和Weng获得的最佳界$0.08n$。这个问题可以看作是关于稠密图三角形分解的Nash-Williams猜想的一个分部模拟，该猜想最近在Delcourt和Postle的突破性工作中被证明。我们的证明使用了一种“放电”策略，改编了Delcourt和Postle的方法，并结合了一种新颖的方法来实现“平衡性”性质，这是分部设置所必需的。
英文摘要
A Latin square of order $n$ is an $n \times n$ array filled with $n$ symbols so that each symbol appears exactly once in every row and column. A partial Latin square of order $n$ is an $n \times n$ array whose cells are either empty or filled in such a way that each symbol appears at most once in every row and column, and at most $n$ distinct symbols are used. In 1983, Daykin and Häggkvist conjectured that every partial Latin square in which each row and column contains at most $n/4$ symbols, and each symbol is used at most $n/4$ times, can be completed to a Latin square. We prove that every partial Latin square in which each row and column contains at most $0.231n$ symbols, and each symbol is used at most $0.231n$ times, can be completed to a Latin square, significantly improving the previous best-known bound of $0.08n$, obtained by Fu and Weng. This problem can be seen as a partite analogue of the Nash-Williams conjecture concerning triangle decompositions of dense graphs, recently proved in the breakthrough work of Delcourt and Postle. Our proof uses a `discharging' strategy, adapting the approach of Delcourt and Postle, combined with a novel method to achieve a `balancedness' property, required for the partite setting.
Comments28 pages
