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Gorenstein and Cohen-Macaulay Matching Complexes

Ashkan Nikseresht

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.

Gorenstein and Cohen-Macaulay Matching Complexes

Abstract

Let be a simple undirected graph. The family of all matchings of forms a simplicial complex called the matching complex of . Here , we give a classification of all graphs with a Gorenstein matching complex. Also we study when the matching complex of 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.

Paper Structure

This paper contains 6 sections, 7 theorems, 1 equation, 3 figures.

Table of Contents

  1. Introduction
  2. Gorenstein Matching Complexes
  3. Cohen-Macaulay Matching Complexes
  4. Cameron-Walker graphs and graphs with large girth
  5. Complete graphs
  6. Acknowledgement

Key Result

Theorem 2.1

Suppose that $H$ is a graph with at least one edge, $G=\mathrm{L}(H)$ and $K$ be an arbitrary field. Then $G$ (or equivalently, the matching complex of $H$) is Gorenstein over $K$ if and only if each connected component of $H$ is either a 5-cycle or a path with length $\leq 2$ or the graph in Figure

Figures (3)

  • Figure 1: A graph whose line graph has a Gorenstein independence complex.
  • Figure 2: Illustrations for the proof of Theorem \ref{['inde gor']}.
  • Figure 3: An example of a Cameron-Walker graph

Theorems & Definitions (15)

  • Theorem 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • proof
  • ...and 5 more