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Let $\Delta$ be a simplicial complex and $\chi$ be an $s$-coloring of
$\Delta$. Biermann and Van Tuyl have introduced the simplicial complex
$\Delta_{\chi}$. As a corollary of Theorems 5 and 7 in their 2013 article,
we obtain that the Stanley--Reisner ring of ${\Delta_{\chi}}$ over a field
is Cohen--Macaulay. In this note, we generalize this corollary by proving
that the Stanley--Reisner ideal of $\Delta_{\chi}$ over a field is
set-theoretic complete intersection. This also generalizes a result of
Macchia.
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