On powers of the cover ideals of graphs

For a simple graph $G$, assume that $J(G)$ is the vertex cover ideal of $G$ and $J(G{)}^{(s)}$ is the $s$-th symbolic power of $J(G)$. We prove that $(J(C{)}^{(s)})=(J(C{)}^s)$ for all $s\ge 1$ and for all odd cycle $C$. For a simplicial complex $\mathrm{\Delta }$, we show that $\mathrm{\Delta }$ is vertex decomposable if $I_{\mathrm{\Delta }}^{\vee }$ is weakly polymatroidal. Let $W=G^{\pi }$ be a fully clique-whiskering graph, we prove that $J(W{)}^s$ is weakly polymatroidal for all $s\ge 1$.