Polynomially reducing NP-Complete problem clarification
Since 3COL is NP-hard and X is clearly in NP, by definition there is some many-one reduction from X to 3COL, though it may be awkward to state. There is a simple oracle reduction along the lines outlined in the other answer: remove each edge at a time, and check if the resulting graph is 3-colorable.

More interesting, perhaps, is a reduction going the other way, thus showing that X is NP-hard (and so NP-complete). Given a graph, we adjoin to it $K_4$, the clique on 4 vertices. The new graph is a Yes instance of X iff the original graph is 3-colorable (exercise).