Chromatic-choosability of hypergraphs with high chromatic number

TL;DR

This paper extends a known graph coloring conjecture to hypergraphs, showing that hypergraphs with high chromatic number and specific size conditions are chromatic-choosable, and identifies sharp bounds for this property.

Contribution

It generalizes Ohba's conjecture to r-uniform hypergraphs and establishes sharp size bounds for chromatic-choosability.

Findings

Hypergraphs with |V(H)| ≤ rχ(H)+r-1 are chromatic-choosable.

Counterexamples exist when |V(H)| = rχ(H)+r.

The conjecture is supported by classes of hypergraphs meeting the size condition.

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 $\chi_l(H)=\chi(H)$.