On the depth of symbolic powers of edge ideals of graphs

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 ${\rm depth} S/(I(G)^{(s)})\geq \alpha_2(G)-s+1$istrueforeverychordalgraph$G$andeveryinteger$s\geq 1$\mathrm{.}Moreover,itisprovedthatforanygraph$G$,wehave${\rm depth} S/(I(G)^{(2)})\geq \alpha_2(G)-1$.