Chromatic equivalence classes of complete tripartite graphs

Some necessary conditions on a graph which has the same chromatic polynomial as the complete tripartite graph Km,n,rKm,n,r are developed. Using these, we obtain the chromatic equivalence classes for Km,n,nKm,n,n (where 1≤m≤n1≤m≤n) and Km1,m2,m3Km1,m2,m3 (where |mi−mj|≤3|mi−mj|≤3). In particular, it is shown that (i) Km,n,nKm,n,n (where 2≤m≤n2≤m≤n) and (ii) Km1,m2,m3Km1,m2,m3 (where |mi−mj|≤3|mi−mj|≤3, 2≤mi,i=1,2,32≤mi,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 Kn1,n2,n3Kn1,n2,n3 J. Systems Sci. Math. Sci. 20 (2000) 181–186].