In k-Digraph Coloring we are given a digraph and are asked to partition its vertices into at most k sets, so that each set induces a DAG. This well-known problem is NP-hard, as it generalizes (undirected) k-Coloring, but becomes trivial if the input digraph is acyclic. This poses the natural parameterized complexity question of what happens when the input is “almost” acyclic. In this paper we study this question using parameters that measure the input’s distance to acyclicity in either the directed or the undirected sense. In the directed sense perhaps the most natural notion of distance to acyclicity is directed feedback vertex set. It is already known that, for all k ≥ 2, k-Digraph Coloring is NP-hard on digraphs of directed feedback vertex set of size at most k + 4. We strengthen this result to show that, for all k ≥ 2, k-Digraph Coloring is already NP-hard for directed feedback vertex set of size exactly k. This immediately provides a dichotomy, as k-Digraph Coloring is trivial if directed feedback vertex set has size at most k − 1. Refining our reduction we obtain three further consequences: (i) 2-Digraph Coloring is NP-hard for oriented graphs of directed feedback vertex set at most 3; (ii) for all k ≥ 2, k-Digraph Coloring is NP-hard for graphs of feedback arc set of size at most k2; interestingly, this leads to a second dichotomy, as we show that the problem is FPT by k if feedback arc set has size at most k2 − 1; (iii) k-Digraph Coloring is NP-hard for graphs of directed feedback vertex k, even if the maximum degree Δ is at most 4k − 1; we show that this is also almost tight, as the problem becomes FPT for digraphs of directed feedback vertex set of size k and Δ ≤ 4k − 3. Since these results imply that the problem is also NP-hard on graphs of bounded directed treewidth, we then consider parameters that measure the distance from acyclicity of the underlying graph. On the positive side, we show that k-Digraph Coloring admits an FPT algorithm parameterized by treewidth, whose parameter dependence is (tw!)ktw. Since this is considerably worse than the ktw dependence of (undirected) k-Coloring, we pose the question of whether the tw! factor can be eliminated. Our main contribution in this part is to settle this question in the negative and show that our algorithm is essentially optimal, even for the much more restricted parameter treedepth and for k = 2. Specifically, we show that an FPT algorithm solving 2-Digraph Coloring with dependence tdo(td) would contradict the ETH. Then, we consider the class of tournaments. It is known that deciding whether a tournament is 2-colorable is NP-complete. We present an algorithm that decides if we can 2-color a tournament in $O^{*}({\sqrt [3]{6}}^{n})$ time. Finally, we explain how this algorithm can be modified to decide if a tournament is k-colorable.

It is well known that every planar or projective planar graph can be 3-colored so that each color class induces a forest. This bound is sharp. In this paper, we show that there are in fact exponentially many 3-colorings of this kind for any (projective) planar graph. The same result holds in the setting of 3-list-colorings.

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.