Bounded powers of edge ideals: regularity and linear quotients

Let $S=K[x_1,\dots ,x_n]$ 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 $(I^k{)}_{\mathfrak{c}}\mathrm{\ne }0$. 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 $s=1,\dots {\delta }_{\mathfrak{c}}(I(G))$, the Castelnuovo–Mumford regularity of $(I(G{)}^s{)}_{\mathfrak{c}}$ is bounded above by ${\delta }_{\mathfrak{c}}(I(G))+s$.