Bounded powers of edge ideals: Gorenstein polytopes

Abstract

Let $S=K[x_1,\dots ,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I(G)\subset S$ the edge ideal of a finite graph $G$ on $n$ vertices. Given a vector $\mathfrak{c}\in {\mathbb{N}}^n$ and an integer $q\ge 1$, we denote by $(I(G{)}^q{)}_{\mathfrak{c}}$ the ideal of $S$ generated by those monomials belonging to $I(G{)}^q$ whose exponent vectors are componentwise bounded above by $\mathfrak{c}$. Let ${\delta }_{\mathfrak{c}}(I(G))$ denote the largest integer $q$ for which $(I(G{)}^q{)}_{\mathfrak{c}}\ne (0)$. Since $(I(G{)}^{{\delta }_{\mathfrak{c}}(I)}{)}_{\mathfrak{c}}$ is a polymatroidal ideal, it follows that its minimal set of monomial generators is the set of bases of a discrete polymatroid $\mathcal{D}(G,\mathfrak{c})$. In the present paper, a classification of Gorenstein polytopes of the form $\mathrm{conv}(\mathcal{D}(G,\mathfrak{c}))$ is studied.