Chromatic equivalence classes of complete tripartite graphs

Some necessary conditions on a graph which has the same chromatic polynomial as the complete tripartite graph K m , n , r are developed. Using these, we obtain the chromatic equivalence classes for K m , n , n (where 1 ≤ m ≤ n ) and K m 1 , m 2 , m 3 (where | m i − m j | ≤ 3 ). In particular, it is shown that (i) K m , n , n (where 2 ≤ m ≤ n ) and (ii) K m 1 , m 2 , m 3 (where | m i − m j | ≤ 3 , 2 ≤ m i , i = 1 , 2 , 3 ) are uniquely determined by their chromatic polynomials. The result (i), proved earlier by Liu et al. [R.Y. Liu, H.X. Zhao, C.Y. Ye, A complete solution to a conjecture on chromatic uniqueness of complete tripartite graphs, Discrete Math. 289 (2004) 175–179], answers a conjecture (raised in [G.L. Chia, B.H. Goh, K.M. Koh, The chromaticity of some families of complete tripartite graphs (In Honour of Prof. Roberto W. Frucht), Sci. Ser. A (1988) 27–37 (special issue)]) in the affirmative, while result (ii) extends a result of Zou [H.W. Zou, On the chromatic uniqueness of complete tripartite graphs K n 1 , n 2 , n 3 J. Systems Sci. Math. Sci. 20 (2000) 181–186].