Chromatic-choosability of hypergraphs with high chromatic number

Abstract: It was conjectured by Ohba and confirmed recently by Noel et al. that, for any graph $G$, if $|V(G)|\le 2\chi(G)+1$ then $\chi_l(G)=\chi(G)$. This indicates that the graphs with high chromatic number are chromatic-choosable. We show that this is also the case for uniform hypergraphs and further propose a generalized version of Ohba's conjecture: for any $r$-uniform hypergraph $H$ with $r\geq 2$, if $|V(H)|\le r\chi(H)+r-1$ then $\chi_l(H)=\chi(H)$. We show that the condition of the proposed conjecture is sharp by giving two classes of $r$-uniform hypergraphs $H$ with $|V(H)|= r\chi(H)+r$ and $\chi_l(H)>\chi(H)$. To support the conjecture, we give two classes of $r$-uniform hypergraphs $H$ with $|V(H)|= r\chi(H)+r-1$ and prove that $\chi_l(H)=\chi(H)$.