We make several observations on the implementation of Edmonds’ blossom algorithm for solving minimum-weight perfectmatching problems and we present computational results for geometric problem instances ranging in size from 1,000 nodes up to 5,000,000 nodes. A key feature in our implementation is the use of multiple search trees with an individual dual-change e for each tree. As a benchmark of t...

In this note, we prove a sufficient condition for the existence of a perfect matching in a regular graph in terms of its eigenvalues and its expansion constant. We improve a recent result of Brouwer and Haemers. French version: Dans cette note, nous prouvons un état suffisant pour l’existence d’un assortiment parfait dans un graphe régulier en termes de ses valeurs propres et son constante d’ex...

Martin Klazar computed the total weight of ordered trees under 12 different notions of weight. The last and perhaps most interesting of these weights, w12, led to a recurrence relation and an identity for which he requested combinatorial explanations. Here we provide such explanations. To do so, we introduce the notion of a “Klazar violator” vertex in an increasing ordered tree and observe that...

Haxell’s condition [Hax95] is a natural hypergraph analog of Hall’s condition, which is a wellknown necessary and sufficient condition for a bipartite graph to admit a perfect matching. That is, when Haxell’s condition holds it forces the existence of a perfect matching in the bipartite hypergraph. Unlike in graphs, however, there is no known polynomial time algorithm to find the hypergraph per...

We characterize the structural impediments to the existence of Borel perfect matchings for acyclic locally countable Borel graphs admitting a Borel selection of finitely many ends from their connected components. In particular, this yields the existence of Borel matchings for such graphs of degree at least three. As a corollary, it follows that acyclic locally countable Borel graphs of degree a...

Altan derivatives of polycyclic conjugated hydrocarbons were recently introduced and studied in theoretical organic chemistry. We now provide a generalization of the altan concept, applicable to any graph. Several earlier noticed topological properties of altan derivatives of polycyclic conjugated hydrocarbons are shown to be the properties of all altan derivatives of all graphs. Among these ar...

For a certain graph obtained by adding extra vertices and edges to the triangular lattice graph, Propp conjectured that the number of perfect matchings of such a graph is always divisible by 3. In this note we prove this conjecture. In a graph G, a matching is a set of edges such that no two edges are incident to each other. A matching in a graph is called perfect if every vertex is incident wi...

We know that counting perfect matchings is polynomial time when we restrict ourselves to the class of planar graphs. Generally speaking, the decision and search versions of a problem turn out to be “easier” than the counting question. For example, the problem of determining if a perfect matching exists, and finding one when it does, is polynomial time in general graphs, while the question of co...

The graph with the largest signless Laplacian spectral radius among all bicyclic graphs with perfect matchings is determined.