Given a graph $G=(V,E)$, for a vertex set $S\subseteq V$, let $N(S)$ denote the set of vertices in $V$ that have a neighbor in $S$. In this paper, we prove the following Hall-type statement. Let $k \ge 2$ be an integer. Let $X$ be a vertex set in the undirected graph $G$ such that for each subset $S$ of $X$ it holds $|N(S)|\ge \frac1k {|S|}$. Then $G$ has a matching of size at least $\frac{|X|}{k+1}$. Using this statement, we derive tight bounds for the estimators of the matching size in planar graphs. These estimators are used in designing sublinear space algorithms for approximating the maching size in the data stream model of computation. In particular, we show the number of locally superior vertices, introduced in \cite{Jowhari23}, is a $3$ factor approximation of the matching size in planar graphs. The previous analysis proved a $3.5$ approximation factor. In another consequence, we show a simple setting of an estimator by Esfandiari \etal \cite{EHLMO15} achieves $3$ factor approximation of the matching size in planar graphs. Namely, let $s$ be the number of edges with both endpoints having degree at most $2$ and let $h$ be the number of vertices with degree at least $3$. We show when the graph is planar, the size of matching is at least $\frac{s+h}3$. This result generalizes a known fact that every planar graph on $n$ vertices with minimum degree $3$ has a matching of size at least $\frac{n}3$.

翻译：给定图$G=(V,E)$，对顶点集$S\subseteq V$，令$N(S)$表示$V$中与$S$存在邻接关系的顶点集合。本文证明如下霍尔型命题：设$k\ge 2$为整数，$X$为无向图$G$中的一个顶点集，满足对$X$的任意子集$S$均有$|N(S)|\ge \frac1k |S|$。则$G$存在大小至少为$\frac{|X|}{k+1}$的匹配。利用该命题，我们推导了平面图中匹配大小估计量的紧致界。这些估计量被用于设计数据流计算模型中近似匹配大小的亚线性空间算法。具体而言，我们证明\cite{Jowhari23}提出的局部优效顶点数是平面图匹配大小的$3$倍近似——此前分析仅得到$3.5$倍近似因子。另一推论表明，Esfandiari等\cite{EHLMO15}提出的简单估计量设置可在平面图中实现$3$倍近似：设$s$为两端点度数均不超过$2$的边数，$h$为度数至少为$3$的顶点数，则当图为平面图时匹配大小至少为$\frac{s+h}3$。该结论推广了已知事实：每个最小度数为$3$的$n$顶点平面图至少包含大小为$\frac{n}3$的匹配。