Hall’s theorem for bipartite graphs gives a necessary and sufficient condition for the existence of a matching in a given bipartite graph. Aharoni and Ziv [2] generalized the notion of matchability to a pair of possibly infinite matroids on the same set and gave a condition that is sufficient for the matchability of a given pair (M,W) of finitary matroids, where the matroid M is SCF — a sum of ...

In this work, we prove a refinement of the Gallai-Edmonds structure theorem for weighted matching polynomials by Ku and Wong. Our proof uses connection between branched continued fractions. We also show how is related to modification Sylvester classical Sturm's on number zeros real polynomial in an interval. addition, obtain some other results about polynomials.

The purpose of this note is to present a proof of Hall's matching theorem (also called marriage theorem) which I have not encountered elsewhere in literature-what yet does not mean that it is necessarily new. We refer to Hall's theorem in the following form: Theorem 1 (Hall). Let n be a positive integer. Let be a bipartite graph whose set of vertices consists of n blue vertices B 1 ; B 2 Some n...

A graph G having a perfect matching (or I-factor) is called n-fextendable if every matching of size n is extended to a I-factor. Further, G is said to be (r : m, n }-extendable if, for every connected subgraph S of order 2r for which G \ V(S) is connected, S is m-extendable and G \ V(S) is nextendable. We prove the following: Let p, r, m, and n be positive integers with p r > nand r > m. Then e...

We obtain generalizations of the Fan’s matching theorem for an open (or closed) covering related to an admissible map. Each of these is restated as a KKM theorem. Finally, applications concerning coincidence theorems and section results are given.

The maximum matching width is a width-parameter that is defined on a branchdecomposition over the vertex set of a graph. The size of a maximum matching in the bipartite graph is used as a cut-function. In this paper, we characterize the graphs of maximum matching width at most 2 using the minor obstruction set. Also, we compute the exact value of the maximum matching width of a grid.

Hall’s Theorem is a basic result in Combinatorics which states that the obvious necesssary condition for a finite family of sets to have a transversal is also sufficient. We present a sufficient (but not necessary) condition on the sizes of the sets in the family and the sizes of their intersections so that a transversal exists. Using this, we prove that in a bipartite graph G (bipartition {A,B...

We consider simple undirected graphs. An edge subset A of G is called an induced n{star packing of G if every component of the subgraph G[A] induced by A is a star with at most n edges and is an induced subgraph of G. We consider the problem of nding an induced n{star packing of G that covers the maximum number of vertices. This problem is a natural generalization of the classical matching prob...