Let B(2m,m) be the set of all bicyclic graphs on 2m(m ≥ 2) vertices with perfect matchings. In this paper, we characterize the bicyclic graphs with minimal number of matchings and maximal number of independent sets in B(2m,m). 2010 Mathematics Subject Classification: 05C69, 05C05

Let G be a graph and let Pm(G) denote the number of perfect matchings of G. We denote the path with m vertices by Pm and the Cartesian product of graphs G and H by G×H. In this paper, as the continuance of our paper [19], we enumerate perfect matchings in a type of Cartesian products of graphs by the Pfaffian method, which was discovered by Kasteleyn. Here are some of our results: 1. Let T be a...

The paper analyzes effects of occupational and regional mobility on the matching rate using the monthly panel disaggregated on regional and occupational level. The main contribution of the paper is measuring the effect of substitutability between vacancies for different occupations and vacancies in different regions on matchings. The estimates indicate higher regional mobility in West Germany b...

We give an upper bound on the number of perfect matchings in an undirected simple graph G with an even number of vertices, in terms of the degrees of all the vertices in G. This bound is sharp if G is a union of complete bipartite graphs. This bound is a generalization of the upper bound on the number of perfect matchings in bipartite graphs on n + n vertices given by the Bregman-Minc inequalit...

The matching preclusion set of a graph is a set of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings. The matching preclusion number is the minimum cardinality over all matching preclusion sets. We show in this paper that, for any ≥  , the matching preclusion numbers of both -dimensional restricted HL-graph and recursive circulant  ...

Lovász and Plummer conjectured, in the mid 1970’s, that every cubic graph G with no cutedge has an exponential in |V (G)| number of perfect matchings. In this work we show that every cubic planar graph G whose geometric dual graph is a stack triangulation has at least 3φ (G)|/72 distinct perfect matchings, where φ is the golden ratio. Our work builds on a novel approach relating Lovász and Plum...

The graphs considered here will be finite and will have no multiple edges. Let G be such a graph. A matching in G is a spanning subgraph whose components are nodes and edges only. We define a k-matching in G to be a matching with k edges. When the matching consists of edges only, it will be called a perfect matching. The number of perfect matchings in G will be denoted by y(G). The total number...

A graph has the neighbour-closed-co-neighbour, or ncc property, if for each of its vertices x, the subgraph induced by the neighbour set of x is isomorphic to the subgraph induced by the closed non-neighbour set of x. Graphs with the ncc property were characterized in [1] by the existence of a locally C4 perfect matching M : every two edges of M induce a subgraph isomorphic to C4. In the presen...

In this work, we investigate in the problem of perfect matchings with prescribed matchings in the n-dimensional hypercube network Qn. We obtain the following contributions: For any arbitrary matching with at most n − 1 edges, it can be extended to a perfect matching of Qn for n ≥ 1. Furthermore, for any arbitrary non-forbidden matching with n edges, it also can be extended to a perfect matching...