The v-Number and Castelnuovo-Mumford Regularity of Cover Ideals of Graphs

Abstract

The $\textrm{v}$-number of a graded ideal $I\subsetneq R$, denoted by $\textrm{v}(I)$, is the minimum degree of a polynomial $f$ for which $I:f$ is a prime ideal. Jaramillo and Villarreal (J Combin Theory Ser A 177:105310, 2021) studied the $\textrm{v}$-number of edge ideals. In this paper, we study the $\textrm{v}$ number of the cover ideal $J(G)$ of a graph $G$. The main result shows that $\textrm{v}(J(G))\leq \textrm{reg}(R/J(G))$ for any simple graph $G$, which is quite surprising because, for the case of edge ideals, this inequality does not hold. Our main result relates $\textrm{v}(J(G))$ with the Cohen-Macaulay property of $R/I(G)$, where $I(G)$ denotes the edge ideal of $G$. We provide an infinite class of connected graphs, which satisfy $\textrm{v}(J(G))=\textrm{reg}(R/J(G))$. Also, we show that for every positive integer $k$, there exists a connected graph $G_{k}$ such that $\textrm{reg}(R/J(G_{k}))-\textrm{v}(J(G_{k}))=k$. Also, we explicitly compute the $\textrm{v}$-number of cover ideals of cycles.