Gorenstein and Cohen-Macaulay Matching Complexes

TL;DR

This paper classifies graphs whose matching complexes are Gorenstein and characterizes when these complexes are Cohen-Macaulay, providing full characterizations for specific graph classes such as those with girth at least 5 or complete graphs.

Contribution

It offers a complete classification of graphs with Gorenstein matching complexes and characterizes Cohen-Macaulay matching complexes in certain graph classes.

Findings

Classified all graphs with Gorenstein matching complexes.

Characterized Cohen-Macaulay matching complexes for graphs with girth ≥ 5.

Fully characterized Cohen-Macaulay matching complexes for complete graphs.

Abstract

Let $H$ be a simple undirected graph. The family of all matchings of $H$ forms a simplicial complex called the matching complex of $H$. Here , we give a classification of all graphs with a Gorenstein matching complex. Also we study when the matching complex of $H$ 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 5 or a complete graph is Cohen-Macaulay.