Regularity, matchings and Cameron-Walker graphs

TL;DR

This paper characterizes when the regularity of the edge ideal of a graph equals its matching number plus one, showing it occurs precisely when each component is a pentagon or Cameron-Walker graph.

Contribution

It provides a complete characterization of graphs for which the regularity equals the matching number plus one, specifically identifying pentagons and Cameron-Walker graphs as the key components.

Findings

Regularity equals matching number plus one for certain graphs.

Connected components are either pentagons or Cameron-Walker graphs.

Provides a characterization criterion for this regularity condition.

Abstract

Let $G$ be a simple graph and let $\nu(G)$ be the matching number of $G$. It is well-known that $\reg I(G) \leqslant \nu(G)+1$. In this paper we show that $\reg I(G) = \nu(G)+1$ if and only if every connected component of $G$ is either a pentagon or a Cameron-Walker graph.