List coloring digraphs

TL;DR

This paper explores the list dichromatic number of digraphs, demonstrating its similarities to the dichromatic number, establishing bounds, and extending classical graph coloring results to directed graphs.

Contribution

It introduces the concept of list dichromatic number for digraphs, proving key properties, bounds, and analogs of graph coloring theorems for directed graphs.

Findings

List dichromatic number matches dichromatic number in small digraphs and tournaments.

Bipartite digraphs can have a list dichromatic number as large as logarithmic in size.

A Brooks-type upper bound is established for digon-free digraphs.

Abstract

The dichromatic number $\vec{\chi}(D)$ of a digraph $D$ is the least number $k$ such that the vertex set of $D$ can be partitioned into $k$ parts each of which induces an acyclic subdigraph. Introduced by Neumann-Lara in 1982, this digraph invariant shares many properties with the usual chromatic number of graphs and can be seen as the natural analog of the graph chromatic number. In this paper, we study the list dichromatic number of digraphs, giving evidence that this notion generalizes the list chromatic number of graphs. We first prove that the list dichromatic number and the dichromatic number behave the same in many contexts, such as in small digraphs (by proving a directed version of Ohba's Conjecture), tournaments, and random digraphs. We then consider bipartite digraphs, and show that their list dichromatic number can be as large as $\Omega(\log_2 n)$. We finally give a Brooks-type upper bound on the list dichromatic number of digon-free digraphs.