On powers of cover ideals of graphs

Abstract

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 $\operatorname{reg}(J(C)^{(s)})=\operatorname{reg}(J(C)^s)$ for all $s\geq 1$ and for all odd cycle $C$. For a simplicial complex $\Delta$, we show that if $I_{\Delta}^{\vee}$ is weakly polymatroidal (not necessarily generated in one degree) then $\Delta$ is vertex decomposable. Some evidences are provided that the converse conclusion of the above result also holds true if $\Delta$ is pure. Let $W=G^{\pi}$ be a fully clique-whiskering graph. We prove that $J(W)^s$ is weakly polymatroidal for all $s\geq 1$.