On chromatic uniqueness of some complete tripartite graphs

Let $P(G, x)$ be a chromatic polynomial of a graph $G$. Two graphs $G$ and $H$ are called chromatically equivalent iff $P(G, x) = H(G, x)$. A graph $G$ is called chromatically unique if $G\simeq H$ for every $H$ chromatically equivalent to $G$. In this paper, the chromatic uniqueness of complete tripartite graphs $K(n_1, n_2, n_3)$ is proved for $n_1 \geqslant n_2 \geqslant n_3 \geqslant 2$ and $n_1 - n_3 \leqslant 5$.