We provide polynomial-time algorithms for counting the number of perfect matchings and the number of matchings in chain graphs, cochain graphs, and threshold graphs. These algorithms are based on newly developed subdivision schemes that we call a recursive decomposition. On the other hand, we show the #P-completeness for counting the number of perfect matchings in chordal graphs, split graphs a...

In this paper we present a pseudo-deterministic RNC algorithm for finding perfect matchings in bipartite graphs. Specifically, our algorithm is a randomized parallel algorithm which uses poly(n) processors, poly(log n) depth, poly(log n) random bits, and outputs for each bipartite input graph a unique perfect matching with high probability. That is, it returns the same matching for almost all r...

Alexey Pokrovskiy Aharoni and Berger conjectured [1] that every bipartite graph which is the union of n matchings of size n + 1 contains a rainbow matching of size n. This conjecture is related to several old conjectures of Ryser, Brualdi, and Stein about transversals in Latin squares. There have been many recent partial results about the Aharoni-Berger Conjecture. When the matchings have size ...

For a bipartite graph G = (V, E), (1) perfect, (2) maximum and (3) maximal matchings are matchings (1) such that all vertices are incident to some matching edges, (2) whose cardinalities are maximum among all matchings, (3) which are contained in no other matching. In this paper, we present three algorithms for enumerating these three types of matchings. Their time complexities are O(|V |) per ...

|This paper describes an algorithm for nding all the perfect matchings in a bipartite graph. By using the binary partitioning method, our algorithm requires O(c(n+m) + n 2:5 ) computational e ort and O(nm) memory storage, (where n denotes the number of vertices, m denotes the number of edges, and c denotes the number of perfect matchings in the given bipartite graph). Keywords|bipartite graph, ...

In the present paper, the minimal proper alternating cycle (MPAC) rotation graph R(G) of perfect matchings of a plane bipartite graph G is defined. We show that an MPAC rotation graph R(G) of G is a directed rooted tree, and thus extend such a result for generalized polyhex graphs to arbitrary plane bipartite graphs. As an immediate result, we describe a one-to-one correspondence between MPAC s...

We analyze the behavior of the Barvinok estimator of the hafnian of even dimension, symmetric matrices with nonnegative entries. We introduce a condition under which the Barvinok estimator achieves subexponential errors, and show that this condition is almost optimal. Using that hafnians count the number of perfect matchings in graphs, we conclude that Barvinok’s estimator gives a polynomialtim...

P.W. Kasteleyn stated that the number of perfect matchings in a graph embedding on a surface of genus g is given by a linear combination of 4 Pfafans of modi ed adjacencymatrices of the graph, but didn't actually give the matrices or the linear combination. We generalize this to enumerating the perfect matchings of a graph embedding on an arbitrary compact boundaryless 2-manifold S with a linea...

The enumeration of perfect matchings of graphs is equivalent to the dimer problem which has applications in statistical physics. A graph G is said to be n-rotation symmetric if the cyclic group of order n is a subgroup of the automorphism group of G. Jockusch (Perfect matchings and perfect squares, J. Combin. Theory Ser. A, 67(1994), 100-115) and Kuperberg (An exploration of the permanent-deter...

The perfect matching polytope of a graph G is the convex hull of the incidence vectors of all perfect matchings in G. We characterise bipartite graphs and near-bipartite graphs whose perfect matching polytopes have diameter 1.