The $\mathrm{v}$-number of Monomial Ideals

TL;DR

This paper explores the properties of the v-number of monomial ideals, especially in relation to graph invariants, polarization, and Cohen-Macaulayness, providing new inequalities and connections to open problems.

Contribution

It generalizes v-number results to arbitrary monomial ideals, relates v-number to graph invariants, and investigates its behavior in various classes of graphs and algebraic properties.

Findings

v-number equals that of the polarization of the ideal

Established inequalities relating v-number, induced matching number, and regularity for specific graph classes

Showed v-number can exceed regularity + 1 for disconnected graphs

Abstract

We generalize some results of $\mathrm{v}$-number for arbitrary monomial ideals by showing that the $\mathrm{v}$-number of an arbitrary monomial ideal is the same as the $\mathrm{v}$-number of its polarization. We prove that the $\mathrm{v}$-number $\mathrm{v}(I(G))$ of the edge ideal $I(G)$, the induced matching number $\mathrm{im}(G)$ and the regularity $\mathrm{reg}(R/I(G))$ of a graph $G$, satisfy $\mathrm{v}(I(G))\leq \mathrm{im}(G)\leq \mathrm{reg}(R/I(G))$, where $G$ is either a bipartite graph, or a $(C_{4},C_{5})$-free vertex decomposable graph, or a whisker graph. There is an open problem in \cite{v}, whether $\mathrm{v}(I)\leq \mathrm{reg}(R/I)+1$ for any square-free monomial ideal $I$. We show that $\mathrm{v}(I(G))>\mathrm{reg}(R/I(G))+1$, for a disconnected graph $G$. We derive some inequalities of $\mathrm{v}$-numbers which may be helpful to answer the above problem for the case of connected graphs. We connect $\mathrm{v}(I(G))$ with an invariant of the line graph $L(G)$ of $G$. For a simple connected graph $G$, we show that $\mathrm{reg}(R/I(G))$ can be arbitrarily larger than $\mathrm{v}(I(G))$. Also, we try to see how the $\mathrm{v}$-number is related to the Cohen-Macaulay property of square-free monomial ideals.