A complete solution to a conjecture on chromatic uniqueness of complete tripartite graphs

Let image be the chromatic polynomial of a graph image. A graph image is chromatically unique if for any graph image, image implies image. Koh, Teo and Chia conjectured that for any integers image and image with image, the complete tripartite graph image is chromatically unique. Let image denote the graph obtained by deleting all edges in image from the complete tripartite image. In this paper, we establish that for any positive integer image, the chromatic equivalence class of image is contained in the family {image and image}. By applying these results, we confirm this conjecture and show that image is chromatically unique if image and image.