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(m1,m2,m3) (where |m(i) - m(j)| <= 3). In particular, it is shown that (i) K(m,n,n) (where 2 <= m <= n) and (ii) K(m1,m2,m3) (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(n1,n2,n3) J. Systems Sci. Math. Sci. 20 (2000) 181-186]. (C) 2008 Elsevier B.V. All rights reserved.