A classical theorem of Gallai states that in every graph that is critical for k-colorings, the vertices of degree k − 1 induce a tree-like graph whose blocks are either complete graphs or cycles of odd length. We provide a generalization to colorings and list colorings of digraphs, where some new phenomena arise. In particular, the problem of list coloring digraphs with the lists at each vertex v having min{d + (v), d − (v)} colors turns out to be NP-hard.

The dichromatic number χ⃗false(Dfalse) 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 article, 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 Ω(prefixlog2n). We finally give a Brooks‐type upper bound on the list dichromatic number of digon‐free digraphs.

It is known (Bollobás [4]; Kostochka and Mazurova [12]) that there exist graphs of maximum degree ∆ and of arbitrarily large girth whose chromatic number is at least c∆/ log ∆. We show an analogous result for digraphs where the chromatic number of a digraph D is defined as the minimum integer k so that V (D) can be partitioned into k acyclic sets, and the girth is the length of the shortest cycle in the corresponding undirected graph. It is also shown, in the same vein as an old result of Erdős [5], that there are digraphs with arbitrarily large chromatic number where every large subset of vertices is 2-colorable.

Brooks' Theorem states that a connected graph G of maximum degree ∆ has chromatic number at most ∆, unless G is an odd cycle or a complete graph. A result of Johansson [6] shows that if G is triangle-free, then the chromatic number drops to O(∆/ log ∆). In this paper, we derive a weak analog for the chromatic number of digraphs. We show that every (loopless) digraph D without directed cycles of length two has chromatic number χ(D) ≤ (1 − e −13 )∆, where∆ is the maximum geometric mean of the out-degree and in-degree of a vertex in D, when∆ is sufficiently large. As a corollary it is proved that there exists an absolute constant α < 1 such that χ(D) ≤ α(∆ + 1) for everỹ ∆ > 2.Keywords: Digraph coloring, dichromatic number, Brooks theorem, digon, sparse digraph. * Research supported by FQRNT (Le Fonds québécois de la recherche sur la nature et les technologies) doctoral scholarship. †