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

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. [ABSTRACT FROM AUTHOR]