Symbolic powers of cover ideal of very well-covered and bipartite graphs

Abstract:

Let $G$ be a graph with $n$ vertices and $S=\mathbb {K}[x_1,\dots ,x_n]$ 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 show that if $G$ is a very well-covered graph such that $J(G)$ has a linear resolution, then for every integer $k\geq 1$, the ideal $J(G)^{(k)}$ has a linear resolution and moreover, the modules $J(G)^{(k)}$ and $S/J(G)^{(k)}$ satisfy Stanley’s inequality, i.e., their Stanley depth is an upper bound for their depth. Finally, we determine a linear upper bound for the Castelnuovo–Mumford regularity of powers of cover ideals of bipartite graphs.