On the list version of a conjecture of Erd\H{o}s and Neumann-Lara

TL;DR

This paper proves the list version of a conjecture relating large list chromatic number of graphs to orientations with large list dichromatic number, extending previous results on chromatic number.

Contribution

It establishes that graphs with high list chromatic number can be oriented to produce digraphs with high list dichromatic number, confirming a list-based version of Erdős and Neumann-Lara's conjecture.

Findings

Graphs with large list chromatic number have orientations with large list dichromatic number.

Every graph of minimum degree d admits an orientation with list dichromatic number at least ln d.

The result extends Alon's theorem from chromatic to list dichromatic number.

Abstract

The dichromatic number of a digraph $D$, denoted by $\vec{\chi}(D)$, is the smallest number of colours required to colour the vertices of $D$ such that each colour class induces an acyclic digraph. A conjecture of Erd\H{o}s and Neumann-Lara states that there exists a function $f(k)$ such that for every graph $G$ with $\chi(G) \geq f(k)$ there is an orientation of $G$ such that the resulting digraph $D$ satisfies $\vec{\chi}(D) \geq k$. We prove the list version of this conjecture: if $G$ has large list chromatic number then there is an orientation of $G$ such that the resulting digraph has large list dichromatic number. The main tool in our result is the following theorem, which is an extension of an analogous result of Alon for the chromatic number: every graph of minimum degree $d$ admits an orientation such that the resulting digraph has list dichromatic number of order at least $\ln d$.