Chromaticity of some families of dense graphs Original Research Article

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 Ḡ is of the form aK1∪bK3∪⋃1⩽i⩽s Pli, 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 H̄ is of the form aK3∪⋃1⩽i⩽s Pui∪⋃1⩽j⩽t 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.