The First Time KE is Broken up

Abstract

A relevant collection is a collection, $F$, of sets, such that each set in $F$ has the same cardinality, $\alpha(F)$. A Konig Egervary (KE) collection is a relevant collection $F$, that satisfies $|\bigcup F|+|\bigcap F|=2\alpha(F)$. An hke (hereditary KE) collection is a relevant collection such that all of his non-empty subsets are KE collections. In \cite{jlm} and \cite{dam}, Jarden, Levit and Mandrescu presented results concerning graphs, that give the motivation for the study of hke collections. In \cite{hke}, Jarden characterize hke collections. Let $\Gamma$ be a relevant collection such that $\Gamma-\{S\}$ is an hke collection, for every $S \in \Gamma$. We study the difference between $|\bigcap \Gamma_1-\bigcup \Gamma_2|$ and $|\bigcap \Gamma_2-\bigcup \Gamma_1|$, where $\{\Gamma_1,\Gamma_2\}$ is a partition of $\Gamma$. We get new characterizations for an hke collection and for a KE graph.