$\mathrm{v}$-number of Lovász-Saks-Schrijver Ideals and (parity) binomial edge ideals of graphs

Abstract

In this paper, we introduce a new framework for computing certain localized $\mathrm{v}$-numbers of a class of ideals called coordinate-saturated ideals, which includes certain classes of Lovász-Saks-Schrijver (LSS) ideals and (generalized) binomial edge ideals associated with graphs. For a forest graph $G$, we derive an explicit formula for the localized $\mathrm{v}$-number of the LSS ideal $L_G^{\mathbb{K}}(d)$, denoted by ${\mathrm{v}}_{{\mathfrak{p}}_{\varnothing }(G)}(L_G^{\mathbb{K}}(d))$, for all $d\ge 2$, where $\mathbb{K}$ is an algebraically closed field. As a consequence, we prove that $\mathrm{v}(L_G^{\mathbb{K}}(d))\le \mathrm{reg}(R/L_G^{\mathbb{K}}(d)),$ where $\mathrm{v}(L_G^{\mathbb{K}}(d))$ and $\mathrm{reg}(R/L_G^{\mathbb{K}}(d))$ denote the $\mathrm{v}$-number of $L_G^{\mathbb{K}}(d)$ and the Castelnuovo-Mumford regularity of $R/L_G^{\mathbb{K}}(d)$, respectively. Also, we give an upper bound for ${\mathrm{v}}_{I_{K_n}}(L_G^{\mathbb{R}}(2))$, where $\mathbb{R}$ is field of real numbers. Furthermore, we provide combinatorial descriptions of the localized $\mathrm{v}$-number of parity binomial edge ideals, denoted by ${\mathrm{v}}_{{\mathfrak{p}}^{+}(G)}({\mathcal{I}}_G)$, and, as an application, show that $\mathrm{v}({\mathcal{I}}_G)\le \mathrm{reg}(R/{\mathcal{I}}_G)$ for several classes of non-bipartite graphs. Finally, we prove that $\mathrm{v}(J_G^k)\le \mathrm{reg}(R/J_G^k)$ for all powers of binomial edge ideals of closed graphs $G$.

40 Pages. Comments are welcome!