Beck et. al. characterized the grid graphs whose perfect matching polytopes are Gorenstein and they also showed that for some parameters, perfect matching polytopes of torus graphs are Gorenstein. In this paper, we complement their result, that is, we characterize the torus graphs whose perfect matching polytopes are Gorenstein. Beck et. al. also gave a method to construct an infinite family of...

In this paper, we study the relationship between forbidden subgraphs and the existence of a matching. Let H be a set of connected graphs, each of which has three or more vertices. A graph G is said to beH-free if no graph inH is an induced subgraph of G. We completely characterize the setH such that every connected H-free graph of sufficiently large even order has a perfect matching in the foll...

Faudree, R.J., R.J. Gould and L.M. Lesniak, Neighborhood conditions and edge-disjoint perfect matchings, Discrete Mathematics 91 (1991) 33-43. A graph G satisfies the neighborhood condition ANC(G) 2 m if, for all pairs of vertices of G, the union of their neighborhoods has at least m vertices. For a fixed positive integer k, let G be a graph of even order n which satisfies the following conditi...

V := {ri}i=1 ∪ {ci}i=1 and E := {{ri, cj} | mi,j ≥ 1/n}. The main observation is that if G contains a perfect matching then there exists a desired permutation. To see that, let us assume that P = {e1, . . . , en} is a subset of the edges of G which form a perfect matching, with ei = {ri, cji}. Then setting π(i) := ji satisfies the desired properties: since P is a matching we have that π is a bi...

Matching extendability is significant in graph theory and its applications. The basic notion in this direction is n-extendability introduced by Plummer in 1980. Motivated by the different natures of bipartite matchings and non-bipartite matchings, this paper investigates bipartite-matching extendable (BM-extendable) graphs. A graph G is said to be BM-extendable if every matching M which is a pe...

the idea of “forcing” has long been used in many research fields, such as colorings, orientations, geodetics and dominating sets in graph theory, as well as latin squares, block designs and steiner systems in combinatorics (see [1] and the references therein). recently, the forcing on perfect matchings has been attracting more researchers attention. a forcing set of m is a subset of m contained...

A near perfect matching is a matching covering all but one vertex in a graph. Let G be a connected graph and n ≤ (|V (G)|−2)/2 be a positive integer. If any n independent edges in G are contained in a near perfect matching, then G is said to be defect n-extendable. This paper presents two construction characterizations of defect n-extendable bipartite graphs and a necessary condition for minima...