Chromatic-Choosability of Hypergraphs with High Chromatic Number

Abstract

It was conjectured by Ohba and confirmed  by Noel, Reed and Wu that, for any graph $G$, if $|V(G)|\le 2\chi(G)+1$ then  $G$ is chromatic-choosable; i.e., it satisfies $\chi_l(G)=\chi(G)$. This indicates that the graphs with high chromatic number are chromatic-choosable. We observe 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  prove that $\chi_l(H)=\chi(H)$ for two classes of  $r$-uniform hypergraphs $H$ with $|V(H)|= r\chi(H)+r-1$.