Assume a very simple graph with 3 points:
V0—V1—V2
The following represent all the different possible colorings using 3 colors. I’ve labelled all of the types of colorings that are isomorphic (is that the right word to apply here)? I’d like to know what the name for the type of coloring that only includes the “unique” colorings. I’ve read some stuff about “canonical” colorings but I’m having trouble parsing it. Is the set for a coloring which contains just C1 and C2 (or {(1,2,1),(1,2,3)}) a “canonical” coloring?
121 - C1
123 - C2
131 - C1 (2->3)
132 - C2 (2->3 and 3->2)
212 - C1 (1->2 and 2->1)
213 - C2 (1->2 and 2->1)
232 - C1 (1->2 and 2->3)
231 - C2 (1->2 and 2->3 and 3->1)
313 - C1 (1->3 and 3->1)
323 - C1 (1->3)
312 - C2 (1->3 and 2->1 and 3->2)
321 - C2 (1->3 and 3->1)
There are two important related terms here that I think you are confusing:
A labelling of a graph is some mapping between the vertices and the integers 1 to N. There are N! different labellings of an 'unlabelled' graph.
Usually a 'coloring' is more general, where different vertices can have the same colors. There is also the idea of a 'proper' coloring where each vertex has neighbours with different colors from itself - or, rather, each edge connects vertices with a different color.
What you have seems to be a mixture between these two - colorings of labelled graphs? So you consider (say) 121 and 131 to be isomorphic? I'm guessing that's what you mean by "(2->3)". If so, then they are in the same 'equivalence class', I suppose. Then you have two such classes, C1 and C2.
A canonical labelling is a single member of the equivalence class of labellings of a graph. For your equivalence classes of colorings, I suppose that the 'minimal' coloring in the class would do as the canonical example for that class