Bounded powers of edge ideals: Gorenstein toric rings

Let S=K[x₁, …,xₙ] denote the polynomial ring in n variables over a field K and I ⊂ S a monomial ideal. Given a vector cⁿ, the ideal Ic is the ideal generated by those monomials belonging to I whose exponent vectors are componentwise bounded above by c. Let δc(I) be the largest integer q for which (Iq)c≠ 0. For a finite graph G, its edge ideal is denoted by $I(G)$. Let B(c,G) be the toric ring which is generated by the monomials belonging to the minimal system of monomial generators of (I(G)^δc(I))c. In a previous work, the authors proved that (I(G)^δc(I))c is a polymatroidal ideal. It follows that B(c,G) is a normal Cohen--Macaulay domain. In this paper, we study the Gorenstein property of B(c,G).