Two nonisomorphic graphs are said to be chromatically equivalent (also termed "chromically equivalent by Bari 1974) if they have identical chromatic
polynomials. A graph that does not share a chromatic
polynomial with any other nonisomorphic graph is said to be a chromatically
unique graph.

The chromatically equivalent simple graphs on five or fewer vertices are illustrated above.

Bari (1974) gives a number of chromatically equivalent graph pairs on 11 to 17 vertices that are planar triangulations.

Bari, R. A. "Chromatically Equivalent Graphs." In Graphs and Combinatorics (Ed. R. A. Bari and F. Harary).
Berlin: Springer-Verlag, pp. 186-200, 1974.Bari, R. A. "Recent
Results on Chromatically Equivalent Graphs." In Second International Conference
on Combinatorial Mathematics (New York, 1978).Ann. New York Acad. Sci.319,
37-46, 1979.