Bounded powers of edge ideals: regularity and linear quotients

Let $S=K[x_1, \ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and let $I \subset S$ be a monomial ideal. For a vector $\mathfrak{c}\in\mathbb{N}^n$, we set $I_{\mathfrak{c}}$ to be the ideal generated by monomials belonging to $I$ whose exponent vectors are componentwise bounded above by $\mathfrak{c}$. Also, let $\delta_{\mathfrak{c}}(I)$ be the largest integer $k$ such that $(I^k)_{\mathfrak{c}}\neq 0$. It is shown that for every graph $G$ with edge ideal $I(G)$, the ideal $(I(G)^{\delta_{\mathfrak{c}}(I)})_{\mathfrak{c}}$ is a polymatroidal ideal. Moreover, we show that for each integer $s=1, \ldots \delta_{\mathfrak{c}}(I(G))$, the Castelnuovo--Mumford regularity of $(I(G)^s)_{\mathfrak{c}}$ is bounded above by $\delta_{\mathfrak{c}}(I(G))+s$.