On the depth of symbolic powers of edge ideals of graphs

Abstract

Assume that $G$ is a graph with edge ideal $I(G)$ and star packing number ${\alpha }_2(G)$. We denote the $s$-th symbolic power of $I(G)$ by $I(G{)}^{(s)}$. It is shown that the inequality $\mathrm{depth}S/(I(G{)}^{(s)})\ge {\alpha }_2(G)-s+1$ is true for every chordal graph $G$ and every integer $s\ge 1$. Moreover, it is proved that for any graph $G$, we have $\mathrm{depth}S/(I(G{)}^{(2)})\ge {\alpha }_2(G)-1$.