Vertex decomposability and regularity of very well-covered graphs

TL;DR

This paper investigates the properties of very well-covered graphs, establishing their Cohen-Macaulay condition via pure vertex decomposability and relating their regularity to the maximum number of pairwise 3-disjoint edges.

Contribution

It proves that very well-covered graphs are Cohen-Macaulay if and only if they are pure vertex decomposable, and relates their regularity to 3-disjoint edges, extending previous results.

Findings

Cohen-Macaulay if and only if pure vertex decomposable for very well-covered graphs

Regularity equals the maximum number of pairwise 3-disjoint edges

Extension of Kummini's result on unmixed bipartite graphs

Abstract

A graph $G$ is well-covered if it has no isolated vertices and all the maximal independent sets have the same cardinality. If furthermore two times this cardinality is equal to $|V(G)|$, the graph $G$ is called very well-covered. The class of very well-covered graphs contains bipartite well-covered graphs. Recently in \cite{CRT} it is shown that a very well-covered graph $G$ is Cohen-Macaulay if and only if it is pure shellable. In this article we improve this result by showing that $G$ is Cohen-Macaulay if and only if it is pure vertex decomposable. In addition, if $I(G)$ denotes the edge ideal of $G$, we show that the Castelnuovo-Mumford regularity of $R/I(G)$ is equal to the maximum number of pairwise 3-disjoint edges of $G$. This improves Kummini's result on unmixed bipartite graphs.