Induced matchings and the v-number of graded ideals

TL;DR

This paper provides a formula for the v-number of graded ideals and explores its bounds in relation to the induced matching number of graphs, with classifications for specific graph classes and insights into W2-graphs.

Contribution

It introduces a formula for the v-number of graded ideals and establishes bounds relating it to the induced matching number for various classes of graphs.

Findings

The v-number of the edge ideal is bounded above by the induced matching number in certain graph classes.

The v-number of the edge ideal is a lower bound for the regularity of the edge ring in these classes.

Classifications are provided for when the bounds are tight for cycles and W2-graphs.

Abstract

We give a formula for the v-number of a graded ideal that can be used to compute this number. Then we show that for the edge ideal $I(G)$ of a graph $G$ the induced matching number of $G$ is an upper bound for the v-number of $I(G)$ when $G$ is very well-covered, or $G$ has a simplicial partition, or $G$ is well-covered connected and contain neither $4$- nor $5$-cycles. In all these cases the v-number of $I(G)$ is a lower bound for the regularity of the edge ring of $G$. We classify when the upper bound holds when $G$ is a cycle, and classify when all vertices of a graph are shedding vertices to gain insight on $W_2$-graphs.