  We study the natural linear operators associated to divide and color (DC)
models. The degree of nonuniqueness of the random partition yielding a DC model
is directly related to the dimension of the kernel of these linear operators.
We determine exactly the dimension of these kernels as well as analyze a
permutation-invariant version. We also obtain properties of the solution set
for certain parameter values which will be important in (1) showing that large
threshold discrete Gaussian free fields are DC models and in (2) analyzing when
the Ising model with a positive external field is a DC model, both in future
work. However, even here, we give an application to the Ising model on a
triangle.
