In this article, Temperley’s bijection between spanning trees of the square grid on the one hand, and perfect matchings (also known as dimer coverings) of the square grid on the other, is extended to the setting of general planar directed (and undirected) graphs, where edges carry nonnegative weights that induce a weighting on the set of spanning trees. We show that the weighted, directed spann...

The structural theory of matchings is used to establish lower bounds on the number of perfect matchings in n-extendable graphs. It is shown that any such graph on p vertices and q edges contains at least (n + 1)!/4[q − p − (n − 1)(2 − 3) + 4] different perfect matchings, where is the maximum degree of a vertex in G. © 2007 Elsevier B.V. All rights reserved. MSC: 05C70; 05C40; 05C75

In this article, Temperley’s bijection between spanning trees of a subgraph of the square grid on the one hand, and perfect matchings (also known as dimer coverings) of an associated subgraph of the square grid on the other, is extended to the setting of general planar directed (and undirected) graphs, where edges carry nonnegative weights that induce a weighting on the set of spanning trees. W...

This paper is concerned with structures of general graphs with perfect matchings. We first reveal a partially ordered structure among factor-components of general graphs with perfect matchings. Our second result is a generalization of Kotzig’s canonical partition to a decomposition of general graphs with perfect matchings. It contains a short proof for the theorem of the canonical partition. Th...

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 prove that the matching graph M(Qd) of the d-dimensional hypercube is bipartite and connected for d ≥ 4. This proves Kreweras’ conjecture [2] that the graph Md is connected, where Md is obtai...

The Berge-Fulkerson Conjecture states that every cubic bridgeless graph has six perfect matchings such that every edge of the graph is in exactly two of the perfect matchings. If the Berge-Fulkerson Conjecture is true, then what can we say about the proportion of edges of a cubic bridgeless graph that can be covered by k of its perfect matchings? This is the question we address in this paper. W...

Graphical condensation is a technique used to prove combinatorial identities among numbers of perfect matchings of plane graphs. Propp and Kuo first applied this technique to prove identities for bipartite graphs. Yan, Yeh, and Zhang later applied graphical condensation to nonbipartite graphs to prove more complex identities. Here we generalize some of the identities of Yan, Yeh, and Zhang. We ...

In this paper we investigate the question whether a perfect matching can be isolated by a weighting scheme using Chinese Remainder Theorem (short: CRT). We give a systematical analysis to a method based on CRT suggested by Agrawal in a CCC’03-paper for testing perfect matchings. We show that this desired test-procedure is based on a deterministic weighting scheme which can be generalized in a n...

Abstract. 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. The enumeration of perfect matchings of graphs with reflective symmetry was studied extensively in the past. In this paper we consider t...