Chromatic Uniqueness of Certain Complete 4-partite Graphs

Let P (G, λ) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P (H, λ) = P (G, λ) implies H is isomorphic to G. Liu et al. [Liu, R. Y., Zhao, H. X., Ye, C. F.: A complete solution to a conjecture on chromatic uniqueness of complete tripartite graphs. Discrete Math., 289, 175–179 (2004)], and Lau and Peng [Lau, G. C., Peng, Y. H.: Chromatic uniqueness of certain complete t-partite graphs. Ars Comb., 92, 353–376 (2009)] show that K(p − k, p − i, p) for i = 0, 1 are chromatically unique if p ≥ k + 2 ≥ 4. In this paper, we show that if 2 ≤ i ≤ 4, the complete tripartite graph K(p− k, p− i, p) is chromatically unique for integers k ≥ i and p ≥ k/4 + i+ 1.