Let P (G,λ) be the chromatic polynomial of a graph G. Two graphs G and H are said to be chromatically equivalent, denoted G ∼ H, if P (G,λ) = P (H,λ). We write [G] = {H|H ∼ G}. If [G] = {G}, then G is said to be chromatically unique. In this paper, we first characterize certain complete 6-partite graphs with 6n + 2 vertices according to the number of 7-independent partitions of G. Using these r...

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...

Let P.G; / be the chromatic polynomial of a graph G. A graph G is chromatically unique if for

We obtain new necessary conditions on a graph which shares the same chromatic polynomial as that of the complete tripartite graph Km,n,r. Using these, we establish the chromatic equivalence classes for K1,n,n+1 (where n ≥ 2). This gives a partial solution to a question raised earlier by the authors. With the same technique, we further show that Kn−3,n,n+1 is chromatically unique if n ≥ 5. In th...

Let P (G,λ) be the chromatic polynomial of a graph G. Two graphs G and H are said to be chromatically equivalent, denoted G ∼ H, if P (G,λ) = P (H,λ). We write [G] = {H|H ∼ G}. If [G] = {G}, then G is said to be chromatically unique. In this paper, we first characterize certain complete 6-partite graphs G with 6n + i vertices for i = 0, 1, 2 according to the number of 7-independent partitions o...

Let P (G,λ) be the chromatic polynomial of a simple graph G. A graph G is chromatically unique if for any simple graph H, P (H,λ) = P (G,λ) implies that H is isomorphic to G. Many sufficient conditions guaranteeing that some certain complete tripartite graphs are chromatically unique were obtained by many scholars. Especially, in 2003, Zou Hui-wen showed that if n > 1 3 m+ 1 3 k+ 1 3 mk+ 1 3 m−...