Depth, Stanley depth and regularity of ideals associated to graphs

Abstract

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots ,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. Let $G$ be a graph with $n$ vertices. Assume that $I=I(G)$ is the edge ideal of $G$ and $J=J(G)$ is its cover ideal. We prove that $\mathrm{sdepth}(J)\ge n-{\nu }_o(G)$ and $\mathrm{sdepth}(S/J)\ge n-{\nu }_o(G)-1$, where ${\nu }_o(G)$ is the ordered matching number of $G$. We also prove the inequalities $\mathrm{sdepth}(J^k)\ge \mathrm{depth}(J^k)$ and $\mathrm{sdepth}(S/J^k)\ge \mathrm{depth}(S/J^k)$, for every integer $k\gg 0$, when $G$ is a bipartite graph. Moreover, we provide an elementary proof for the known inequality $\mathrm{reg}(S/I)\le {\nu }_o(G)$.