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On the list version of a conjecture of Erdős and Neumann-Lara

Ararat Harutyunyan, Lucas Picasarri-Arrieta, Gil Puig i Surroca

Abstract

The dichromatic number of a digraph $D$, denoted by $\vecχ(D)$, 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 Erdős and Neumann-Lara states that there exists a function $f(k)$ such that for every graph $G$ with $χ(G) \geq f(k)$ there is an orientation of $G$ such that the resulting digraph $D$ satisfies $\vecχ(D) \geq k$. 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 $\ln d$.

On the list version of a conjecture of Erdős and Neumann-Lara

Abstract

The dichromatic number of a digraph , denoted by , is the smallest number of colours required to colour the vertices of such that each colour class induces an acyclic digraph. A conjecture of Erdős and Neumann-Lara states that there exists a function such that for every graph with there is an orientation of such that the resulting digraph satisfies . We prove the list version of this conjecture: if has large list chromatic number then there is an orientation of 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 admits an orientation such that the resulting digraph has list dichromatic number of order at least .
Paper Structure (3 sections, 10 theorems, 18 equations)

This paper contains 3 sections, 10 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The proof

Key Result

Theorem 3

Every graph $G$ with minimum degree $d$ satisfies $\vec{\chi}_{\ell}(G) \geqslant (\tfrac{1}{3}-o(1))\log_2 d$.

Theorems & Definitions (20)

  • Conjecture 1: erdosPNCN1979
  • Conjecture 2: puigTHESIS
  • Theorem 3
  • Corollary 4
  • proof
  • Theorem 5: Rambaud et al. perso_rambaud
  • proof
  • Lemma 6: Chernoff
  • Lemma 7: manberSJC13
  • Lemma 8: charArXiv25
  • ...and 10 more