arXiv 1603.065522016-03-27math.CO

Konig Egervary系列

Hereditary Konig Egervary Collections

Adi Jarden

备注：: 20 Pages

摘要：

设$G$是一个顶点集为$V(G)$的简单图.如果$S$中没有两个顶点相邻，则$V(G)$的子集$S$是独立的。如果图$G$满足$α(G) + μ(G)= |V(G)|$，则称图$G$是Konig-Egervary（简称KE）图，其中$α(G)$表示最大独立集的大小，$μ(G)$表示最大匹配的基数.令$Ω(G)$表示所有最大独立集的族。集合集合$F$是一个类集合，如果$|\bigcup Γ|+|\bigcap Γ|=2α$对$F$的每个子集合$Γ$成立。我们的特点是一个hke收集和调用新的特征的KE图。本文证明了图$G$的存在唯一性，使得$Ω(G)$是一个极大hke集合。它是一个二分图。从而解决了Jarden，Levit和Mandrescu的\cite{jlm}问题，证明了对于某个图$G$和$|\bigcup F|+|\bigcap F|=2α(F)$，$F$是一个hke集合当且仅当它是$Ω(G)$的一个子集.最后，我们证明了具有$α(F)=α$和$|\bigcup F|=n$的类集合$F$的最大基数是$2^{n-α}$。

英文摘要：

Let $G$ be a simple graph with vertex set $V(G)$. A subset $S$ of $V(G)$ is independent if no two vertices from $S$ are adjacent. The graph $G$ is known to be a Konig-Egervary (KE in short) graph if $α(G) + μ(G)= |V(G)|$, where $α(G)$ denotes the size of a maximum independent set and $μ(G)$ is the cardinality of a maximum matching. Let $Ω(G)$ denote the family of all maximum independent sets. A collection $F$ of sets is an hke collection if $|\bigcup Γ|+|\bigcap Γ|=2α$ holds for every subcollection $Γ$ of $F$. We characterize an hke collection and invoke new characterizations of a KE graph. We prove the existence and uniqueness of a graph $G$ such that $Ω(G)$ is a maximal hke collection. It is a bipartite graph. As a result, we solve a problem of Jarden, Levit and Mandrescu \cite{jlm}, proving that $F$ is an hke collection if and only if it is a subset of $Ω(G)$ for some graph $G$ and $|\bigcup F|+|\bigcap F|=2α(F)$. Finally, we show that the maximal cardinality of an hke collection $F$ with $α(F)=α$ and $|\bigcup F|=n$ is $2^{n-α}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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