4-color to 3-color polynomial reduction
For a more general reduction, László Lovász [1] proved that

COLORABILITY reduces to 3-COLORABILITY

The COLORABILITY problem is defined as follows:

Input: A graph $G$ and a positive integer $k$.
Problem: Is $G$ $k$-colorable?

That is, the reduction given by László Lovász is applicable to any $k$ ($k \ge 4$).
However, this reduction is quite intricate. You can find an excellent explanation by Vašek Chvátal here. Please free feel to ask if you have some difficulty in understanding this reduction.

[1] Coverings and coloring of hypergraphs, Proceedings of the Fourth Southeastern Conference on Combinatorics, Graph Theory, and Computing, Utilitas Math., Winnipeg, Man., 1973.