## “School of Mathematics”

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Paper   IPM / M / 17631
 School of Mathematics Title: A Hall-type theorem with algorithmic consequences in planar graphs Author(s): Hossein Jowhari (Joint with E. Ghorbani) Status: Published Journal: Discrete Math. Vol.: 347 Year: 2024 Pages: 113892 Supported by: IPM
Abstract:
Given a graph G =(V, E), for a vertex set SÃ¢??V, let N(S) denote the set of vertices in V that have a neighbor in S. Extending the concept of binding number of graphs by Woodall (1973), for a vertex set XÃ¢??V, we define the binding number of X, denoted by bind(X), as the maximum number such that for every SÃ¢??X where N(S) != V(G) it holds that |N(S)| Ã¢?Â¥b|S|. Given this definition, we prove that if a graph V(G) contains a subset X with bind(X) =1/k where k is an integer, then G possesses a matching of size at least |X|/(k +1). Using this statement, we derive tight bounds for the estimators of the matching size in planar graphs. These estimators are previously used in designing sublinear space algorithms for approximating the matching size in the data stream model of computation. In particular, we show that the number of locally superior vertices is a 3factor approximation of the matching size in planar graphs. The previous analysis by Jowhari (2023) proved a 3.5approximation factor. As another application, we show a simple variant of an estimator by Esfandiari et al. (2015) achieves 3factor approximation of the matching size in planar graphs. Namely, let s be the number of edges with both endpoints having degree at most 2and let h be the number of vertices with degree at least 3. We prove that when the graph is planar, the size of matching is at least (s +h)/3. This result generalizes a known fact that every planar graph on n vertices with minimum degree 3has a matching of size at least n/3.