  Let $P(G, x)$ be the chromatic polynomial of a graph $G$. A graph $G$ is
called \textit{chromatically unique} if for any graph $H,\, P(G, x) = P(H, x)$
implies that $G$ and $H$ are isomorphic. In this paper we show that full
tripartite graph $K(n_1, n_2, n_3)$ is chromatically unique if $n_1 \geq n_2
\geq n_2 \geq n_3 \geq 2, n_1 - n_3 \leq 5$ and $n_1 + n_2 + n_3 \not \equiv 2
\mod{3}$.
