We determine the exact minimum l-degree threshold for perfect matchings in k-uniform hypergraphs when the corresponding threshold for perfect fractional matchings is significantly less than 1 2 ( n k−l ) . This extends our previous results [18, 19] that determine the minimum l-degree thresholds for perfect matchings in k-uniform hypergraphs for all l ≥ k/2 and provides two new (exact) threshold...

Using elements of the structural theory of matchings and a recently proved conjecture concerning bricks, it is shown that every n-extendable brick (except K4, C6 and the Petersen graph) with p vertices and q edges contains at least q − p + (n − 1)!! perfect matchings. If the girth of such an n-extendable brick is at least five, then this graph has at least q − p + nn−1 perfect matchings. As a c...

Aslam presents an algorithm he claims will count the number of perfect matchings in any incomplete bipartite graph with an algorithm in the function-computing version of NC, which is itself a subset of FP. Counting perfect matchings is known to be #P-complete; therefore if Aslam's algorithm is correct, then NP=P. However, we show that Aslam's algorithm does not correctly count the number of per...

In the last decade there have been many results about special families of graphs whose number of perfect matchings is given by perfect or near perfect powers (N. Elkies et al., J. Algebraic Combin. 1 (1992), 111– 132; B.-Y. Yang, Ph.D. thesis, Department of Mathematics, MIT, Cambridge, MA, 1991; J. Propp, New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999). In this pa...

In 1961, P.W. Kasteleyn enumerated the domino tilings of a 2n × 2n chessboard. His answer was always a square or double a square (we call such a number "squarish"), but he did not provide a combinatorial explanation for this. In the present thesis, we prove by mostly combinatorial arguments that the number of matchings of a large class of graphs with 4-fold rotational symmetry is squarish; our ...

We consider families of \(k\)-subsets \(\{1, \dots, n\}\), where \(n\) is a multiple \(k\), which have no perfect matching. An equivalent condition for family \(\mathcal{F}\) to matching there be blocking set, set \(b\) elements n\}\) that cannot covered by disjoint sets in \(\mathcal{F}\). are specifically interested the largest possible size with and less than \(b\). Frankl resolved case sing...

Given an edge-weighted graph G, let PerfMatch(G) denote the weighted sum over all perfect matchings M in G, weighting each matching M by the product of weights of edges in M. If G is unweighted, this plainly counts the perfect matchings of G. In this paper, we introduce parity separation, a new method for reducing PerfMatch to unweighted instances: For graphs G with edge-weights 1 and −1, we co...

A graph G with a perfect matching is called saturated if G + e has more perfect matchings than G for any edge e that is not in G. Lovász gave a characterization of the saturated graphs called the cathedral theorem, with some applications to the enumeration problem of perfect matchings, and later Szigeti gave another proof. In this paper, we give a new proof with our preceding works which reveal...

If G is a bridgeless cubic graph, Fulkerson conjectured that we can find 6 perfect matchings M1, . . . ,M6 of G with the property that every edge of G is contained in exactly two of them and Berge conjectured that its edge set can be covered by 5 perfect matchings. We define τ(G) as the least number of perfect matchings allowing to cover the edge set of a bridgeless cubic graph and we study thi...

perfect matchings. (A perfect matching or 1-factor is a set of disjoint edges covering all vertices.) This generalizes a result of Voorhoeve [11] for the case k = 3, stating that any 3-regular bipartite graph with 2n vertices has at least ( 4 3) n perfect matchings. The base in (1) is best possible for any k: let αk be the largest real number such that any k-regular bipartite graph with 2n vert...