If I modify a 3-COL problem (with colors A, B, C) in a way that I now demand that at most, say, 50 vertices may be colored with color A, does the problem still remain NP-complete?
I've been thinking of a reduction from a standard 3-COL, but there is no way that I can solve a standard 3-colorability using the problem described above; I'd have to cut the graph into smaller pieces. Other NP-complete problems I know of, like independence set, have a parameter and in my problem there is a fixed number instead of a parameter. So this got me thinking, that maybe such a problem could be solved in P-time, but this doesn't make sense as I could probably replace 50 with any other number.
So this is simpler than asking whether there is a 3-COL such that at most k vertices receive color A, but still quite hard in general I think.
Am I correct?