Coloring vertices in a cubic lattice graph and counting edges between vertices of identical and vertices of distinct coloration Take an $A \times B \times C$ cubic lattice graph $G$, and paint $k_1$ vertices with color $c_1$ & $k_2$ vertices with color $c_2$, where $(k_1 + k_2)$ is equal to the total vertex count.  Let $s_1$ be the number of edges between vertices of identical coloration, and $s_2$ be the number of edges between vertices of distinct coloration.  How many pairs $(s_1, s_2)$ exist?  
Also, is it known how many distinct total colorations of $G$, with $k_1$ and $k_2$ colors of type $c_1$ and $c_2$, respectively, exist up to rotational & reflectional symmetry of the graph?
Edit: I need to specify that, in the above problem, $k_1$ and $k_2$ are exactly specified, and I am looking for an exact counting solution as a function of $A$, $B$, and $C$ (i.e. as a function of the edge lengths of the cubic lattice). I apologize for any confusion this may have caused.
Edit 2: Actually I would also be perfectly happy with an exact counting solution where $A$, $B$, and $C$ are exactly specified, and we have an exact counting solution as a function of $k_1$ and $k_2 = (A*B*C) - k_1$.
 A: This is more a collection of potentially useful ideas and intuitions, with no guarantee of
correctness and no proof.
If you take a coloring and tweak it by switching the colors on two vertices of opposite color,
you land in the same class of colorings $(k_1,k_2)$ while making an incremental change in the
s values of any number from 0 up to twice the maximal degree D in the graph, since you have vertex
degrees ranging over all values in [D/2,D].  I expect the s values to range over a large interval, missing
at most D values near the ends of the interval, and only when $k_1$ or $k_2$ is smaller than D.
Using a checkerboard coloring, $s_2$ can be pushed toward $Dk_1$ until you run out of colors or edges.
Using a compact coloring, one can push $s_2$ down to AB + A + 1 or slightly lower depending on
$k_1$ being a large enough multiple of A or AB.  When $k_1$ is smaller than A a different analysis will
be needed.  I assume A<  B< C. (So it may be that A is the limiting factor more than D in the
previous paragraph.)
I think that this can be reframed in terms of general bipartite graphs and can benefit
from research on numerical semigroups and the Frobenius coin problem.  In particular,
I do not see constraints that indicate large exceptional values for $s_2$.
