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- W2949841034 abstract "A graph $G$ is called emph{chromatic-choosable} if its choice number is equal to its chromatic number, namely $Ch(G)=chi(G)$. Ohba has conjectured that every graph $G$ satisfying $|V(G)|leq 2chi(G)+1$ 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: $K_{3*2,2*(k-3),1}$, $K_{3,2*(k-1)}$, $K_{s+3,2*(k-s-1),1*s}$, $K_{4,3,2*(k-4),1*2}$, and $K_{5,3,2*(k-5),1*3}$. 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 $Ch(K_{3*3,2*(k-5),1*2})=k$ and $Ch(K_{4,3*2,2*(k-6),1*3})=k$ (for $kgeq 5$ and $kgeq 6$, respectively)." @default.
- W2949841034 created "2019-06-27" @default.
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- W2949841034 date "2013-08-14" @default.
- W2949841034 modified "2023-10-18" @default.
- W2949841034 title "Choice number of complete multipartite graphs $K_{3*3,2*(k-5),1*2}$ and $K_{4,3*2,2*(k-6),1*3}$" @default.
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- W2949841034 doi "https://doi.org/10.48550/arxiv.1308.3046" @default.
- W2949841034 hasPublicationYear "2013" @default.
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