A Hall-type theorem with algorithmic consequences in planar graphs

Given a graph G = (V, E), for a vertex set S subset of 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 X subset of V, we define the binding number of X, denoted by bind(X), as the maximum number b such that for every S subset of X where N(S) not equal V(G) it holds that vertical bar N(S)vertical bar >= b vertical bar S vertical bar. 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 vertical bar X vertical bar/(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 3 factor approximation of the matching size in planar graphs. The previous analysis by Jowhari (2023) proved a 3.5 approximation 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 2 and 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 3 has a matching of size at least n/3. (c) 2024 Elsevier B.V. All rights reserved.