Chromaticity of some families of dense graphs

For a graph G, let P(G; ) be its chromatic polynomial and let [G] be the set of graphs having P(G; ) 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 4rst determine [G] for each graph G whose complement 5 G is of the form aK1∪bK3∪⋃16i6s Pli , where a; b are any nonnegative integers and li is even. By this result, we 4nd that such a graph G is chromatically unique i7 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 5 H is of the form aK3 ∪⋃16i6s Pui ∪⋃16j6t Cvj , where ui ¿ 3 and ui ≡ 4 (mod 5) 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. c © 2002 Elsevier Science B.V. All rights reserved.