A Characterization of Two-Dimensional Buchsbaum Matching Complexes

The matching complex $M(G)$ of a graph G is the set of all matchings in G. A Buchsbaum simplicial complex is a generalization of both a homology manifold and a Cohen–Macaulay complex. We give a complete characterization of the graphs G for which $M(G)$ is a two-dimensional Buchsbaum complex. As an intermediate step, we determine which graphs have matching complexes that are themselves connected graphs.