A Note on Chromatic Uniqueness of Completely Tripartite Graphs

Let P (G,λ) be the chromatic polynomial of a simple graph G. A graph G is chromatically unique if for any simple graph H, P (H,λ) = P (G,λ) 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 > 1 3 m+ 1 3 k+ 1 3 mk+ 1 3 m− 1 3 k+ 2 3 √ m2 + k2 +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 χ–unique). In this paper, we prove that for any non-negative integers n,m and k, where m ≥ 2 and k ≥ 0, if n ≥ 1 3 m + 1 3 k + 1 3 mk + 1 3 m− 1 3 k + 4 3 , then the complete tripartite graph K(n − m,n, n + k) is χ–unique, which is an improvement on Zou Hui-wen’s result in the case m ≥ 2 and k ≥ 0. Furthermore, we present a related conjecture.