  In \cite{st2017}, the authors introduced the so-called \emph{general divide
and color models}. One of the most well-known examples of such a model is the
Ising model with external field $h = 0$, which has a color representation given
by the random cluster model. In this paper, we give necessary and sufficient
conditions for this color representation to be unique. We also show that if one
considers the Ising model on a complete graph, then for many $h > 0$, there is
no color representation. This shows in particular that any generalization of
the random cluster model which provides color representations of Ising models
with external fields cannot, in general, be a generalized divide and color
models. Furthermore, we show that there can be at most finitely many $\beta >
0$ at which the random cluster model can be continuously extended to a color
representation for $h \not = 0$.
