About chromatic uniqueness of some complete tripartite graphs

Abstract

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 $n_1 + n_2 + n_3 \not \equiv 2 \mod{3}$.