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.
Edit: With "general" I mean: considering in a systematic way different "natural" and "important"coloring functions (not all of them at once), studying their properties and relating them in insightful ways.
(Compare this to general graph theory: The general graph is just a function $V\times V \rightarrow \{0,1\}$ and at that level of generality one might doubt that there's much of a theory. But there is.)