# General theory of graph coloring

In Ben Steven's article Colored graphs and their properties I read:

We "color" a graph by assigning various colors to the vertices of that graph. [...] this process of coloring is generally governed by a set of coloring rules. For example, the most basic set of coloring rules, referred to as regular coloring, consists of a single rule: no two adjacent vertices may have the same color.

What I am looking for is a truly general theory of graph colorings and resp. general coloring rules. The theory should be so general to include symmetric (= orderless) context-free grammars.

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Where do the context-free grammars enter into it? – MJD Jul 5 '12 at 17:33
Via the grammatical roles (= colors) the grammatical constituents (= symbols of the alphabet) do play. – Hans Stricker Jul 5 '12 at 17:42
Any hint for downvoting is welcome. – Hans Stricker Jul 5 '12 at 20:39
It wasn't me. ${}{}$ – MJD Jul 5 '12 at 20:42

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A coloring is a function from the vertices of a (finite) graph to an initial segment of the natural numbers. A coloring rule is a subset of the set of all such functions. At that level of generality, I doubt there's much of a theory. – Gerry Myerson Jul 6 '12 at 1:52
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