In this paper it is shovm that every maximum matching in a 3-connectccl graph, other than ](4, contains at least one contractible edge. In the case of a. perfect ma.tching, those graphs in \vhich there exists a perfect matching containing precisely one contractible edge arc characterized.

A graph is called matching covered if for its every edge there is a maximum matching containing it. It is shown that minimal matching covered graphs contain a perfect matching.

Let X,Y be finite sets, r,s,h,??N with s?r,X?Y. By ?Xh we mean the collection of all h-subsets X where each subset occurs ? times. A coloring (partition) is r-regular if element in exactly r subsets color. one-regular color class a perfect matching. We are interested necessary and sufficient conditions under which an can embedded into s-regular ?Yh. Using algebraic techniques involving glueing ...

We will present a new method, which enables us to find threshold functions for many properties in random intersection graphs. This method will be used to establish sharp threshold functions in random intersection graphs for k–connectivity, perfect matching containment and Hamilton cycle containment. keywords: random intersection graph, threshold functions, connectivity, Hamilton cycle, perfect ...

The anti-Kekulé number is the smallest number of edges that must be removed from a connected graph with a perfect matching so that the graph remains connected, but has no perfect matching. In this paper the values of the Anti-Kekulé numbers of the infinite triangular, rectangular and hexagonal grids are found, and they are, respectively, 9, 6 and 4.

In this article, a class of architecture design problems is explored with perfect matchings (PMs). A perfect matching in a graph is a set of edges such that every vertex is present in exactly one edge. The perfect matching approach has many desirable properties such as complete design space coverage. Improving on the pure perfect matching approach, a tree search algorithm is developed that more...

This paper proves that the complexity of exact weight perfect matching problem is NPcomplete by reduction from the good perfect matching problem. Following this result, the other two open problems DNA sequence analysis and discrete min-max assignment problems are proven to be NPcomplete.

We define the anti-forcing number of a perfect matching M of a graph G as the minimal number of edges of Gwhose deletion results in a subgraph with a unique perfect matching M , denoted by af (G,M). The anti-forcing number of a graph proposed by Vukičević and Trinajstić in Kekulé structures of molecular graphs is in fact the minimum anti-forcing number of perfect matchings. For plane bipartite ...

A graph G is a (d, d + 1)-graph if the degree of each vertex of G is either d or d + 1. If d ≥ 2 is an integer and G a (d, d + 1)-graph with exactly one odd component and with no almost perfect matching, then we show in this paper that |V (G)| ≥ 4(d + 1) + 1 and |V (G)| ≥ 4(d + 1) + 3 when d is odd. This result generalizes corresponding statements by C. Zhao (J. Combin. Math. Combin. Comput. 9 ...

Kreweras’ conjecture [1] asserts that every perfect matching of the hypercube Qd can be extended to a Hamiltonian cycle. We [2] proved this conjecture but here we present a simplified proof. The matching graph M(G) of a graph G has a vertex set of all perfect matchings of G, with two vertices being adjacent whenever the union of the corresponding perfect matchings forms a Hamiltonian cycle. We ...