A Hall-type theorem with algorithmic consequences in planar graphs

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 $\frac{n}3$.