Bounded powers of edge ideals: Gorenstein polytopes
Takayuki Hibi, Seyed Amin Seyed Fakhari
TL;DR
The paper develops a framework to classify Gorenstein lattice polytopes arising as conv\(\mathcal{D}(G,\mathfrak{c})\) attached to edge ideals of finite graphs via discrete polymatroids. Central to the approach is the ground-set rank function \(\rho_{(G,\mathfrak{c})}\) and a criterion that links Gorenstein-ness to a simple fractional condition on closed, inseparable subsets. The authors give comprehensive classifications for several graph families (complete graphs, cycles, complete bipartite graphs, paths, regular bipartite graphs, whiskered graphs, and Cameron--Walker graphs), revealing that many Gorenstein polytopes are standard forms such as the unit cube relatives \( {\mathcal{Q}}_n \) and \( {\mathcal{Q}}'_n+(1,...,1) \), while also identifying exceptional cases. Overall, the work connects combinatorial graph theory, discrete polymatroids, and toric-geometry-style polytopes to illuminate when these convex hulls attain the Gorenstein property and how they relate to reflexive and exceptional polytopes.
Abstract
Let $S=K[x_1, \ldots,x_n]$ 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 $q\geq 1$, 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 $δ_{\mathfrak{c}}(I(G))$ denote the largest integer $q$ for which $(I(G)^q)_{\mathfrak{c}}\neq (0)$. Since $(I(G)^{δ_{\mathfrak{c}}(I)})_{\mathfrak{c}}$ 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 ${\rm conv}(\mathcal{D}(G,\mathfrak{c}))$ is studied.