The dichromatic number of a digraph is the minimum integer $k$ such that it admits a $k$-dicolouring, i.e. a partition of its vertices into $k$ acyclic subdigraphs. We say that a digraph $D$ is a super-orientation of an undirected graph $G$ if $G$ is the underlying graph of $D$. If $D$ does not contain any pair of symmetric arcs, we just say that $D$ is an orientation of $G$. In this work, we give both lower and upper bounds on the dichromatic number of super-orientations of chordal graphs. We also show a family of orientations of cographs for which the dichromatic number is equal to the clique number of the underlying graph.

翻译：有向图的二色数定义为最小的整数$k$，使得该图允许一个$k$-二染色，即将其顶点划分为$k$个无圈子图。我们称有向图$D$是无向图$G$的超定向，如果$G$是$D$的基础图。若$D$不包含任何对称弧对，则仅称$D$为$G$的定向。本文中，我们给出了弦图超定向的二色数的上下界。同时，我们展示了一类余图的定向，其二色数等于基础图的团数。