Depth and Stanley depth of symbolic powers of cover ideals of graphs

Let $G$ be a graph with $n$ vertices and let $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over a field $\mathbb{K}$. Assume that $J(G)$ is the cover ideal of $G$ and $J(G)^{(k)}$ is its $k$-th symbolic power. We prove that the sequences $\{{\rm sdepth}(S/J(G)^{(k)})\}_{k=1}^\infty$ and $\{{\rm sdepth}(J(G)^{(k)})\}_{k=1}^\infty$ are non-increasing and hence convergent. Suppose that $\nu_{o}(G)$ denotes the ordered matching number of $G$. We show that for every integer $k\geq 2\nu_{o}(G)-1$, the modules $J(G)^{(k)}$ and $S/J(G)^{(k)}$ satisfy the Stanley's inequality. We also provide an alternative proof for \cite[Theorem 3.4]{hktt} which states that ${\rm depth}(S/J(G)^{(k)})=n-\nu_{o}(G)-1$, for every integer $k\geq 2\nu_{o}(G)-1$.