Choice number of complete multipartite graphs image and image

A graph image is called chromatic-choosable if its choice number is equal to its chromatic number, namely image. Ohba has conjectured that every graph image satisfying image is chromatic-choosable. Since each image-chromatic graph is a subgraph of a complete image-partite graph, we see that Ohba’s conjecture is true if and only if it is true for every complete multipartite graph. However, the only complete multipartite graphs for which Ohba’s conjecture has been verified are: image, image, image, image, and image. In this paper, we show that Ohba’s conjecture is true for two new classes of complete multipartite graphs: graphs with three parts of size 3 and graphs with one part of size 4 and two parts of size 3. Namely, we prove that image and image (for image and image, respectively).