Some Chromatic Equivalence Classes of Complete Multipartite Graphs

Let P(G, lambda) be the chromatic polynomial of a graph G. For graphs G and H, if P(G, lambda) = P(H, lambda), then G and H are said to be chromatically equivalent. A graph G is chromatically unique if for any graph H, P(H, lambda) = P(G, lambda) implies H is isomorphic to G. Let [G] {H | P(H, lambda) = P(G, lambda)} be the chromatic equivalence class determined by G. It is well known that a complete t-partite graph K(1, (p2),...(Pt)) is chromatically unique if and only if p(i) <= 2 (2 <= i <= t). However, the chromatic equivalence class of K(1, p, q) for q >= 3 is only known for K(1,1, q), K(1,2, 4), K(1, 3,4) and K(1, q, q). In this paper, we completely determined the chromatic equivalence class of K(1, p-1, p) for p >= 3 which partially answered the question "What is the chromatic equivalence class for the graph K(1, p, q), where 2 <= p < q ?" raised in [Discrete Maths. 309 (2008), 134-143]. We also completely determined the chromatic equivalence class of K(1, p,...,p).