Coloured Configurations

Consider graphs built with the units $A$:
and $B$: , where the units are glued along
the vertical edges as in the graph .
A configuration of type $(a, b, c)$ is a graph thus built of $a$ units $A$ and $b$ units $B$, where the graph’s vertices are coloured using up to $c$ colours, so that no two adjacent vertices have the same colour.
The compound graph above is an example of a configuration of type $(2,2,6)$, in fact of type $(2,2,c)$ for all $c \ge 4$.
Let $N(a, b, c)$ be the number of configurations of type $(a, b, c)$.
For example, $N(1,0,3) = 24$, $N(0,2,4) = 92928$ and $N(2,2,3) = 20736$.
Find the last $8$ digits of $N(25,75,1984)$.

This problem is an interesting one because it combines elements of combinatorics (counting structures) and graph coloring, a topic in graph theory. It requires high-level mathematical concepts and understanding, usually at the graduate level.

However, without all necessary mathematical formulas or diagrams that should initially be provided (as these are clearly missing in your problem due to formatting issues), and without a known functional relationship between the variables they give, it’s impossible to arrive at a concrete numerical answer.

The problem refers to a configuration by indicating units $A$ and $B$ but there is no associated information or diagram. The functions $N(1,0,3)$, $N(0,2,4)$, and $N(2,2,3)$ also lack supporting equations or mathematical expressions to guide the solution.

Also, the solving process of such complex problems usually can’t be done by a conversational AI yet without having the necessary algorithms on how it will organize and compute for the data.

The proper way to approach such problem for humans would involve creating a model for the given conditions and developing or applying counting procedures that regard the coloring conditions.

Solving large values like $N(25,75,1984)$ would require a systematic approach or formula based on the definition of the configurations and more advanced techniques as it is clearly related to the concept of colorings on planar graphs.

Once a formula is established the calculation of $N(25,75,1984)$ can be done. Given it will likely be a large number, one might then apply modular arithmetic to find the last $8$ digits.

Yet, unfortunately, this is beyond conversational AI capabilities to perform these steps. It is recommended to seek out a specialist in combinatorics or graph theory to help with such problems.

More Answers:
Prize Strings
Best Approximations
Squarefree Numbers

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