Can 3-coloring be reduced to 3-clique? I'm a slight disagreement with my professor over whether or not a certain reduction is possible. He asked us to reduce the problem of 3-Coloring to the problem of 3-Clique. The problem is that I'm fairly confident that 3-Coloring is NP-Complete, while 3-Clique is P. Correct me if I'm wrong (which is very likely), but for any k-clique where the k is fixed, is $V^k$, meaning the 3-clique is $V^3$, right? I asked my professor about this and his response was: 

"3-clique is definitely not in P.  You (apparently) have to examine all thrices of vertices to settle the matter."

And I still don't understand how this is not a $V^3$ operation. 

If I figured out a way to reduce 3-coloring to 3-clique wouldn't I be millionaire?
You are correct. 

3-clique can be solved in $O(n^3)$ time, whereas 3-coloring is NP-hard.

So there can be no "poly-time reduction" from 3-coloring to 3-clique, unless $P=NP$.