Geodesic
On choosability of complete multipartite graphs $K_{4,3*t,2*(k-2t-2),1*(t+1)}$
Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 2, pp. 237-244.

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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(K_4,3*t,2*(k-2t-2),1*(t+1)) = k, which extends the results ch(K_4,3,2*(k-4),1*2) = k given by Shen et al. (Discrete Math. 308 (2008) 136-143), and ch(K_4,3*2,2*(k-6),1*3) = k given by He et al. (Discrete Math. 308 (2008) 5871-5877).
Keywords: list coloring, complete multipartite graphs, chromatic-choosable graphs, Ohba's conjecture 
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Zheng, Guo-Ping; Shen, Yu-Fa; Chen, Zuo-Li; Lv, Jin-Feng. On choosability of complete multipartite graphs $K_{4,3*t,2*(k-2t-2),1*(t+1)}$. Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 2, pp. 237-244. http://geodesic.mathdoc.fr/item/DMGT_2010_30_2_a3/

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