The dichromatic number \(\vec {\chi }(D)\) of a digraph D is the minimum integer k such that D 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. In general, the dichromatic number of such digraphs is bounded above by the clique number of the underlying graph (because chordal graphs are perfect). However, this bound can be improved when we restrict the symmetric part of such a digraph. Let \(D=(V,A)\) be a super-orientation of a chordal graph G. Let B(D) be the undirected graph with vertex set V in which uv is an edge if and only if both uv and vu belongs to A. An easy greedy procedure shows \(\vec {\chi }(D) \leqslant \Bigg \lceil \frac{\omega (G) + \Delta (B(D))}{2} \Bigg \rceil \) . We show that this bound is best possible by constructing, for every fixed \(k,\ell \) with \(k\geqslant \ell +1\) , a super-orientation \(D_{k,\ell }\) of a chordal graph \(G_{k,\ell }\) such that \(\omega (G_{k,\ell }) = k\) , \(\Delta (B(D_{k,\ell })) = \ell \) and \(\vec {\chi }(D_{k,\ell }) = \big \lceil \frac{k+\ell }{2}\big \rceil \) . When \(\Delta (B(D)) = 0\) (i.e. D is an orientation of G), we give another construction showing that this is tight even for orientations of interval graphs. Next, we show that \(\vec {\chi }(D) \leqslant \frac{1}{2}\omega (G) + O(\sqrt{d \cdot \omega (G)})\) with d the maximum average degree of B(D). Finally, we show that if B(D) contains no \(C_4\) as a subgraph, then \(\vec {\chi }(D) \leqslant \bigg \lceil \frac{\omega (G)+3}{2} \bigg \rceil \) . We justify that this is almost best possible by constructing, for every fixed k, a super-orientation \(D_{k}\) of a chordal graph \(G_{k}\) with clique number k such that \(B(D_k)\) is a disjoint union of paths and \(\vec {\chi }(D_k) = \left\lfloor \frac{k+3}{2} \right\rfloor \) . We also exhibit a family of orientations of cographs for which the dichromatic number is equal to the clique number of the underlying graph.
