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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 $K_{4,3*t,2*(k-2t-2),1*(t+1)}$ for all integers t ≥ 1 and k ≥ 2t+2, that is, $ch\left(K_{4,3*t,2*(k-2t-2),1*(t+1)}\right)=k$, which extends the results $ch\left(K_{4,3,2*(k-4),1*2}\right)=k$ given by Shen et al. (Discrete Math. 308 (2008) 136-143), and $ch\left(K_{4,3*2,2*(k-6),1*3}\right)=k$ given by He...