The $\mathrm{v}$-number and Castelnuovo-Mumford regularity of cover ideals of graphs

Abstract: The $\mathrm{v}$-number of a graded ideal $I\subseteq R$, denoted by $\mathrm{v}(I)$, 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 $\mathrm{v}$-number of edge ideals. In this paper, we study the $\mathrm{v}$-number of the cover ideal $J(G)$ of a graph $G$. The main result shows that $\mathrm{v}(J(G))\leq \mathrm{reg}(R/J(G))$ 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 $\mathrm{v}(J(G))$ with the Cohen-Macaulay property of $R/I(G)$. We provide an infinite class of connected graphs, which satisfy $\mathrm{v}(J(G))=\mathrm{reg}(R/J(G))$. Also, we show that for every positive integer $k$, there exists a connected graph $G_k$ such that $\mathrm{reg}(R/J(G_k))-\mathrm{v}(J(G_k))=k$. Also, we explicitly compute the $\mathrm{v}$-number of cover ideals of cycles.