Improvement of the numerical condition of chromaticity on complete tripartite graphs

Let P(G, λ) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P(H, λ)=P(G, λ) implies G≌H. By comparing the number of the triangular subgraph and that of the quadrangular subgraph without chords, the chromatic uniqueness on the tripartite graph K(n-k,n-v,n) was discussed. It was proved that K(n-k,n-v,n) is chromatically unique for n≥v2(k-v/3)/4+v and k≥v≥2 and that K(n-k,n-2,n) is chromatically unique for n≥k+2,k≥2.