A theory of graph coloring

A while back I was posting Huffman trees onto my 20 line Ncurses display.  They ended up filling the console window with the nodes of the tree spread proper;ly, relative to each other, on the plain grid.

Consider the banker who has a Huffman tree of depositors. Put that on my 20 line Ncurses grid, and also put the loan tree on the same grid. The bankers job is to make the market by filling in her chits, on either graph, such that the covered grid is 'white', the two are spectral matched, each equally imprecise in sampling the other.  Neither will meet the Shannon sampling theorem.  But they will have a bounded mismatch if they are packing sphere somewhere.

We can show this puzzle breaks down to the standard vertex coloring problem of graphs.  Our solution implies an equal density, or consistent density over the grid.  The two colored vertex patterns will nearly balance, ad their exits an extended graph with links between opposite colors, almost. We work the problem backwards, we find the combined graph that  expresses a two coloring, almost.  This is like Taylor approximation in graphs, I can do it because Huffman codes have spacial density property, the more precise the tree, the more dense, and density spread as this is a balanced queuing process, a sphere packer.

I think if we scale and round we can fit them both to the same grid and derive interest charges from the scale while returning the round back to individual accounts. Going to three color, I am not sure the process is commutative, I ain't thunk it yet.

The total subject, regarding central banking treasuries unmatched with private sector depositors. Off white is a cycle in graph coloring business, mis coloring in a graph is a loop shorter than the number of colors. It is spectrally off white meaning the grid size aint't matching and you have sample aliasing, as in sample theory. That is the very definition of a cycle.