The dichromatic number of a directed graph is the smallest number of colors needed to color its vertices so that every color class induces an acyclic digraph. The condition on color classes is equivalent to requiring that no directed cycle be monochromatic (Neumann-Lara 1982).
A nonempty acyclic digraph has dichromatic number 1, while a directed cycle has dichromatic number 2. For a tournament, each color class can be ordered so that all its arcs point forward. Replacing every edge of an undirected graph by two oppositely directed arcs makes its dichromatic number equal to the original chromatic number.
For an oriented graph 
, the dichromatic number also equals the minimum of 
over all vertex
orders 
,
where 
has an edge for each arc pointing backward in that
order (Aboulker et al. 2026). Replacing 
in this formula by the clique
number gives the directed clique number.
