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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. 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