Gorenstein and Cohen–Macaulay matching complexes

Abstract

Let [Formula: see text] be a simple undirected graph. The family of all matchings of [Formula: see text] forms a simplicial complex called the matching complex of [Formula: see text]. Here, we give a classification of all graphs with a Gorenstein matching complex. Also we study when the matching complex of [Formula: see text] is Cohen–Macaulay and, in certain classes of graphs, we fully characterize those graphs which have a Cohen–Macaulay matching complex. In particular, we characterize when the matching complex of a graph with girth at least five or a complete graph is Cohen–Macaulay.