For a graph G, let P(G,λ) be its chromatic polynomial and let [G] be the set of graphs having P(G,λ) as their chromatic polynomial. We call [G] the chromatic equivalence class of G. If [G]={G}, then G is said to be chromatically unique. In this paper, we first determine [G] for each graph G whose complement is of the form , where a,b are any nonnegative integers and li is even. By this result, we find that such a graph G is chromatically unique iff ab=0 and li≠4 for all i. This settles the conjecture that the complement of Pn is chromatically unique for each even n with n≠4. We also determine [H] for each graph H whose complement is of the form , where ui⩾3 and for all i. We prove that such a graph H is chromatically unique if ui+1≠vj for all i,j and ui is even when ui⩾6.