Combinatorial characterizations of the Cohen-Macaulayness of the second power of edge ideals

TL;DR

This paper provides combinatorial criteria to determine when the second powers of edge ideals of graphs are Cohen-Macaulay, Buchsbaum, or generalized Cohen-Macaulay, with a classification for bipartite graphs.

Contribution

It offers necessary and sufficient combinatorial conditions for the Cohen-Macaulayness of second powers of edge ideals and classifies bipartite graphs with these properties.

Findings

Characterization of graphs with Cohen-Macaulay second powers

Classification of bipartite graphs with Cohen-Macaulay second powers

Conditions for Buchsbaum and generalized Cohen-Macaulay properties

Abstract

Let $I(G)$ be the edge ideal of a simple graph $G$. In this paper, we will give sufficient and necessary combinatorial conditions of $G$ in which the second symbolic and ordinary power of its edge ideal are Cohen-Macaulay (resp. Buchsbaum, generalized Cohen-Macaulay). As an application of our results, we will classify all bipartite graphs in which the second (symbolic) powers are Cohen-Macaulay (resp. Buchsbaum, generalized Cohen-Macaulay).