Let $S=K[x_1, \ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I \subset S$ a monomial ideal. Given a vector $\mathfrak{ c}\in\mathbb{ N}^n$, the ideal $I_{ \mathfrak{ c}}$ is the ideal generated by those monomials belonging to $I$ whose exponent vectors are componentwise bounded above by $\mathfrak{ c}$. Let $\delta_{ \mathfrak{ c}}(I)$ be the largest integer $q$ for which $(I^q)_{ \mathfrak{ c}}\neq 0$. For a finite graph $G$, its edge ideal is denoted by $I(G)$. Let $\mathcal{ B}(\mathfrak{ c},G)$ be the toric ring which is generated by the monomials belonging to the minimal system of monomial generators of $(I(G)^{ \delta_{ \mathfrak{ c}}(I)})_{ \mathfrak{ c}}$. In a previous work, the authors proved that $(I(G)^{ \delta_{ \mathfrak{ c}}(I)})_{ \mathfrak{ c}}$ is a polymatroidal ideal. It follows that $\mathcal{ B}(\mathfrak{ c},G)$ is a normal Cohen--Macaulay domain. In this paper, we study the Gorenstein property of $\mathcal{ B}(\mathfrak{ c},G)$.
Categories: math.AC math.CO