A Note on Chromatic Uniqueness of Completely Tripartite Graphs

Abstract:

      Let $P(G,\lambda)$ be the chromatic polynomial of a simple graph $G.$ A graph $G$ is chromatically unique if for any simple graph $H,$ $P(H,\lambda)=P(G,\lambda)$ implies that $H$ is isomorphic to $G$. Many sufficient conditions guaranteeing that some certain complete tripartite graphs are chromatically unique were obtained by many scholars. Especially, in 2003, Zou Hui-wen showed that if $n>\frac{1}{3}m^{2} \frac{1}{3}k^{2} \frac{1}{3}mk \frac{1}{3}m-\frac{1}{3}k \frac{2}{3} \sqrt{m^{2} k^{2} mk}$, where $n,k$ and $m$ are non-negative integers, then the complete tripartite graph $K(n-m,n,n k)$ is chromatically unique (or simply $\chi$--unique). In this paper, we prove that for any non-negative integers $n, m$ and $k,$ where $m\geq2$ and $k\geq0,$ if $n\geq\frac{1}{3}m^{2} \frac{1}{3}k^{2} \frac{1}{3}mk \frac{1}{3}m-\frac{1}{3}k \frac{4}{3}$, then the complete tripartite graph $K(n-m,n,n k)$ is $\chi$--unique, which is an improvement on Zou Hui-wen's result in the case $m\geq2$ and $k\geq0.$ Furthermore, we present a related conjecture.