Bounded powers of edge ideals: The strong exchange property

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 $\mathfrak{c}\in\mathbb{Z}_{>0}^n$, 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 $(I^q)_{\mathfrak{c}}\neq 0$. Let $I(G) \subset S$ denote the edge ideal of a finite graph $G$ on the vertex set $V(G) = \{x_1, \ldots, x_s\}$. In our previous work, it is shown that $(I(G)^{\delta_{\mathfrak{c}}(I)})_{\mathfrak{c}}$ is a polymatroidal ideal. Let $\mathcal{W}(\mathfrak{c},G)$ denote the minimal system of monomial generators of $(I(G)^{\delta_{\mathfrak{c}}(I)})_{\mathfrak{c}}$. It follows that $\mathcal{W}(\mathfrak{c},G)$ satisfies the symmetric exchange property. In the present paper, the question when $\mathcal{W}(\mathfrak{c},G)$ enjoys the strong exchange property, or equivalently, when $\mathcal{W}(\mathfrak{c},G)$ is of Veronese type is studied.