AI 中文总结
研究均匀和几乎均匀的3 - 定性独立超图，建立与合并约翰逊图的结构对应，重点是核。针对3 - QI(8,2)分类强独立集并确定其强独立数，证明其为核，还给出n > 8时相关超图是核的充分条件。
AI 中文摘要
定性独立超图为分析覆盖阵列的存在性和结构提供了一个有用的组合框架。在这项工作中，我们研究均匀和几乎均匀的3 - 定性独立超图3 - UQI(n,2)和3 - AUQI(n,2)，并在这些族与合并约翰逊图之间建立结构对应，重点是它们的核。针对最小未解决实例3 - QI(8,2)，我们对其所有强独立集进行分类并确定其强独立数。利用此以及其强色数和最大3 - 团的大小，我们证明3 - QI(8,2)是一个核。对于n > 8，我们进一步确定3 - UQI(n,2)和3 - AUQI(n,2)是核的充分条件。
英文摘要
Qualitative independence hypergraphs provide a useful combinatorial framework for analyzing the existence and structure of covering arrays. In this work, we study the \emph{uniform} and \emph{almost-uniform $3$-qualitative independence hypergraphs} $3\text{-}UQI(n,2)$ and $3\text{-}AUQI(n,2)$, and establish a structural correspondence between these families and merged Johnson graphs, with emphasis on their cores. Focusing on the smallest unresolved instance, $3\text{-}QI(8,2)$, we classify all of its strongly independent sets and determine its strong independence number. Using this, along with its strong chromatic number and the size of the largest $3$-clique, we show that $3\text{-}QI(8,2)$ is a core. For $n>8$, we further identify sufficient conditions under which $3\text{-}UQI(n,2)$ and $3\text{-}AUQI(n,2)$ are cores.
