Abstract
The dichromatic number of a digraph $D$ is the minimum number of colors needed to color its vertices in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has become the focus of numerous works. In this work we look at possible extensions of the Gyárfás-Sumner conjecture. In particular, we conjecture a simple characterization of sets $\mathcal F$ of three digraphs such that every digraph with sufficiently large dichromatic number must contain a member of $\mathcal F$ as an induced subdigraph. 
 Among notable results, we prove that oriented $K_4$-free graphs without a directed path of length $3$ have bounded dichromatic number where a bound of $414$ is provided. We also show that an orientation of a complete multipartite graph with no directed triangle is $2$-colorable. To prove these results we introduce the notion of nice sets that might be of independent interest.
Highlights
Despite the fact that the chromatic number of graphs is arguably the most studied invariant in graph theory, there are still many questions about chromatic number for which we do not have a satisfying answer
A lot of work has been done about the following question: what induced substructures are expected to be found inside a graph if we assume it has very large chromatic number? Or equivalently what are the minimal families F such that the class of graphs that do not contain any graph in F as an induced subgraph has bounded chromatic number? Since complete graphs have unbounded chromatic number and do not contain any induced subgraph other than complete graphs themselves, it is clear that such an F must contain a complete graph
Given a digraph D on k vertices we associate with it a 3-edge-colored complete graph D whose vertices are the vertices of D, and whose edges are colored as follows: for each pair {x, y} of vertices that induce no arc in D, xy is a red edge in D, if the pair induces a K2, xy is a blues edge of D and if it induces a K2, xy is a green edge
Summary
Despite the fact that the chromatic number of graphs is arguably the most studied invariant in graph theory, there are still many questions about chromatic number for which we do not have a satisfying answer. The dichromatic number of a digraph D, denoted χ(D), is the minimum number of colors needed to color the vertices of D in such a way that no directed cycle is monochromatic. In other words, it is the minimum number of acyclic induced subdigraphs needed to partition V (D). Given a class F of digraphs we denote by F orbind(F ) the set of digraphs which have no member of F as an induced subdigraph. What are the finite sets F of digraphs for which the class F orbind(F ) has bounded dichromatic number?
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