A characterization of two-dimensional Buchsbaum matching complexes

TL;DR

This paper characterizes exactly which graphs have two-dimensional Buchsbaum matching complexes, linking graph properties with topological and algebraic features of their matching complexes.

Contribution

It provides a complete classification of graphs whose matching complexes are two-dimensional Buchsbaum complexes, advancing understanding of their topological structure.

Findings

Identifies graphs with two-dimensional Buchsbaum matching complexes

Determines which graphs have matching complexes that are connected graphs

Establishes a link between graph properties and Buchsbaum topological features

Abstract

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.