Choice number of complete multipartite graphs K3∗3,2∗(k−5),1∗2 and K4,3∗2,2∗(k−6),1∗3

AbstractA graph G is called chromatic-choosable if its choice number is equal to its chromatic number, namely Ch(G)=χ(G). Ohba has conjectured that every graph G satisfying |V(G)|≤2χ(G)+1 is chromatic-choosable. Since each k-chromatic graph is a subgraph of a complete k-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: K3∗2,2∗(k−3),1, K3,2∗(k−1), Ks+3,2∗(k−s−1),1∗s, K4,3,2∗(k−4),1∗2, and K5,3,2∗(k−5),1∗3. 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 Ch(K3∗3,2∗(k−5),1∗2)=k and Ch(K4,3∗2,2∗(k−6),1∗3)=k (for k≥5 and k≥6, respectively)