# List colouring of graphs with at most $\big(2-o(1)\big)\chi$ vertices

@article{Reed2003ListCO, title={List colouring of graphs with at most \$\big(2-o(1)\big)\chi\$ vertices}, author={Bruce A. Reed and Benny Sudakov}, journal={arXiv: Combinatorics}, year={2003} }

Ohba has conjectured \cite{ohb} that if the graph $G$ has $2\chi(G)+1$ or fewer vertices then the list chromatic number and chromatic number of $G$ are equal. In this paper we prove that this conjecture is asymptotically correct. More precisely we obtain that for any $0<\epsilon<1$, there exist an $n_0=n_0(\epsilon)$ such that the list chromatic number of $G$ equals its chromatic number, provided $$n_0 \leq |V(G) | \le (2-\epsilon)\chi(G).$$

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