A Note on Divisibility of the Number of Matchings of a Family of Graphs

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 with an edge of the matching. In particular, graphs on an odd number of vertices have no perfect matchings. Many different problems of matchings have been studied: existence, construction, and enumeration are three big categories of problems involving matchings.