We find recursive formulae for the number of perfect matchings in a graph G by splitting G into subgraphs H and Q. We use these formulas to count perfect matching of P hypercube Qn. We also apply our formulas to prove that the number of perfect matching in an edge-transitive graph is , where denotes the number of perfect matchings in G, is the graph constructed from by deleting edges with an en...

we find recursive formulae for the number of perfect matchings in a graph g by splitting g into subgraphs h and q. we use these formulas to count perfect matching of p hypercube qn. we also apply our formulas to prove that the number of perfect matching in an edge-transitive graph is , where denotes the number of perfect matchings in g, is the graph constructed from by deleting edges with an en...

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...

We refer the reader to Jerrum's book [1] for the analysis of a Markov chain for generating a random matching of an arbitrary graph. Here we'll look at how to extend the argument to sample perfect matchings in dense graphs and arbitrary bipartite graphs. Both of these results are due to Jerrum and Sinclair [2]. Dense graphs We'll need some definitions and notations first: Definition 7.1 A graph ...

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...

We introduce a family of graphs, called cellular, and consider the problem of enumerating their perfect matchings. We prove that the number of perfect matchings of a cellular graph equals a power of 2 tiroes the number of perfect matchings of a certain subgraph, called the core of the graph. This yields, as a special case, a new proof of the fact that the Aztec diamond graph of order n introduc...

The perfect matching problem is known to be in ¶, in randomized NC, and it is hard for NL. Whether the perfect matching problem is in NC is one of the most prominent open questions in complexity theory regarding parallel computations. Grigoriev and Karpinski [GK87] studied the perfect matching problem for bipartite graphs with polynomially bounded permanent. They showed that for such bipartite ...

Abstract For sets of n points, even, in general position the plane, we consider straight-line drawings perfect matchings on them. It is well known that such admit at least $$C_{n/2}$$ C n / 2 different plane match...