Three quesition about graph coloring



X(G):= chromatic number of simple graph G


Define a function CLRS: {graphs} -> { sets of function(s)}
s.t. G |-> {f | f:V(G)->{1,2,...,X(G)} is proper coloring}


Suppose G, H be any simple graph...
Is "CLRS(G)=CLRS(H) <-> G is isomophic to H " ?
Is always exist simple graph F s.t. CLRS(F)=CLRS(G)∩CLRS(H)?
Is always exist simple graph F s.t. CLRS(F)=CLRS(G)∪CLRS(H)?

At first, I never know the discussion for set of coloring.
I just get the idea, but I'm not ensure whether it's good or bad.
I just get a new thinking about this.

Is any book ever talk about something like this?

Thank you.

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