On choosability of complete multipartite graphs ${\displaystyle K_{4,3\cdot t,2\cdot (k-2t-2),1\cdot (t+1)}}$

A graph G is said to be chromatic-choosable if ch(G) = χ(G). Ohba has conjectured that every graph G with 2χ(G)+1 or fewer vertices is chromatic-choosable. It is clear that Ohba's conjecture is true if and only if it is true for complete multipartite graphs. In this paper we show that Ohba's conjecture is true for complete multipartite graphs ${\displaystyle K_{4,3\cdot t,2\cdot (k-2t-2),1\cdot (t+1)}}$ for all integers t ≥ 1 and k ≥ 2t+2, that is, ${\displaystyle ch\left(K_{4,3\cdot t,2\cdot (k-2t-2),1\cdot (t+1)}\right)=k}$, which extends the results ${\displaystyle ch\left(K_{4,3,2\cdot (k-4),1\cdot 2}\right)=k}$ given by Shen et al. (Discrete Math. 308 (2008) 136-143), and ${\displaystyle ch\left(K_{4,3\cdot 2,2\cdot (k-6),1\cdot 3}\right)=k}$ given by He et al. (Discrete Math. 308 (2008) 5871-5877).