On the Cores of Uniform and Almost-Uniform $3$-Qualitative Independence Hypergraphs

Abstract

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.