We consider the problem of counting matchings in planar graphs. While perfect matchings in planar graphs can be counted by a classical polynomial-time algorithm [26, 33, 27], the problem of counting all matchings (possibly containing unmatched vertices, also known as defects) is known to be #P-complete on planar graphs [23]. To interpolate between the hard case of counting matchings and the eas...

The method of graphical vertex-condensation for enumerating perfect matchings of plane bipartite graph was found by Propp (Theoret. Comput. Sci. 303(2003), 267-301), and was generalized by Kuo (Theoret. Comput. Sci. 319 (2004), 29-57) and Yan and Zhang (J. Combin. Theory Ser. A, 110(2005), 113125). In this paper, by a purely combinatorial method some explicit identities on graphical vertex-cond...

We determine the exact minimum `-degree threshold for perfect matchings in kuniform hypergraphs when the corresponding threshold for perfect fractional matchings is significantly less than 12 ( n k−` ) . This extends our previous results that determine the minimum `-degree thresholds for perfect matchings in k-uniform hypergraphs for all ` > k/2 and provides two new (exact) thresholds: (k, `) =...

We investigate the existence of a group action τ that is homomesic with respect to alignments, a type of statistic in perfect matchings. Homomesy is defined as the consistency of an average, and perfect matchings are defined as the set of all partitions of 1 to 2n into pairs. We take advantage of the bijection between labeled Dyck paths and perfect matchings to investigate to investigate the po...

In this paper we give a proof that the largest set of perfect matchings, in which any two contain a common edge, is the set of all perfect matchings that contain a fixed edge. This is a version of the famous Erdős-Ko-Rado theorem for perfect matchings. The proof given in this paper is algebraic, we first determine the least eigenvalue of the perfect matching derangement graph and use properties...

The size and number of maximum matchings in a network have found a large variety of applications in many fields. As a ubiquitous property of diverse real systems, power-law degree distribution was shown to have a profound influence on size of maximum matchings in scale-free networks, where the size of maximum matchings is small and a perfect matching often does not exist. In this paper, we stud...

The Hungarian method is an e cient algorithm for nding a minimal cost perfect matching in a weighted bipartite graph. This paper describes an e cient algorithm for nding all minimal cost perfect matchings. The computational time required to generate each additional perfect matching is O(n(n + m)); and it requires O(n+m) memory storage. This problem can be solved by algorithms for nding the Kth-...