定性独立超图$3\text{-}QI(11,2)$的强着色
Strong Colouring of the Qualitative Independence Hypergraph $3\text{-}QI(11,2)$
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中文总结 AI 辅助
研究定性独立超图$3\text{-}QI(11,2)$的强着色，通过将顶点视为特定子集并评估相交集系统确定强独立数，利用相关界确定覆盖阵列数为11并给出超图满足此条件的充分条件。
中文摘要 AI 辅助
我们使用一种技术确定定性独立超图$3\text{-}QI(11, 2)$的强独立数，该技术将其顶点视为$\{1,2, \ldots, 11\}$的子集并作为相交集系统进行评估。这给出了任何强着色中颜色类的最大大小，从而得到$3\text{-}QI(11,2)$强色数的下界。我们利用此下界以及$3\text{-}QI(10,2)$强色数的上界，确定$3\text{-}QI(11, 2)$的覆盖阵列数$CAN(3\text{-}QI(11,2),2) = 11$，并给出超图$H$满足$CAN(H, 2)=11$的充分条件。
英文摘要
We determine the strong independence number of the qualitative independence hypergraph, $3\text{-}QI(11, 2)$, using a technique that involves considering its vertices as subsets of $\{1,2, \ldots, 11\}$ and assessing them as intersecting set systems. This gives the maximum size of colour classes in any strong colouring and thus, a lower bound on the strong chromatic number of $3\text{-}QI(11,2)$. We leverage this bound along with an upper bound of the strong chromatic number of $3\text{-}QI(10,2)$, to consequently, establish that the covering array number of $3\text{-}QI(11, 2)$, $CAN(3\text{-}QI(11,2),2) = 11$ and give a sufficient condition for a hypergraph $H$ to have $CAN(H, 2)=11$.
