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 \((I^q)_{\mathfrak {c}}\ne 0\) . 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 \(\mathcal {B}(\mathfrak {c},G)\) .
