Résumé
The dichromatic number 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 chi ->(D)<=omega(G)+Delta(B(D))2. We show that this bound is best possible by constructing, for every fixed k,l with k >= l+1, a super-orientation Dk,l of a chordal graph Gk,l such that omega(Gk,l)=k, Delta(B(Dk,l))=l and chi ->(Dk,l)=k+l2. 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 chi ->(D)<= 12 omega(G)+O(d<middle dot>omega(G)) with d the maximum average degree of B(D). Finally, we show that if B(D) contains no C4 as a subgraph, then chi ->(D)<=omega(G)+32. We justify that this is almost best possible by constructing, for every fixed k, a super-orientation Dk of a chordal graph Gk with clique number k such that B(Dk) is a disjoint union of paths and chi ->(Dk)=k+32. We also exhibit a family of orientations of cographs for which the dichromatic number is equal to the clique number of the underlying graph.