Abstract
This thesis focuses on a notion of colouring of digraphs introduced by ErdH{o}s and Neumann-Lara in the late 1970s, namely the dicolouring, and its associated digraph parameter: the dichromatic number. It appears in the last decades that many classical results on proper graph colouring have directed counterparts using these notions.We first give a collection of bounds on the dichromatic number of digraphs for which the underlying graph is chordal. We then strengthen the Directed Brooks' Theorem on a large class of digraphs which contains digraphs without antiparallel arcs. We also introduce a notion of variable degeneracy for digraphs which leads to a more general version of this theorem.Next we prove a collection of results on k-dicritical digraphs, that is the digraphs that are minimal obstruction for the (k-1)-dicolourability. We first generalise a result of Gallai to the directed case and then prove a conjecture of Kostochka and Stiebitz in the particular case k=4. We also discuss the maximum density of such digraphs and prove that the number of 3-dicritical semi-complete digraphs is finite. We then give a collection of results on the substructures in large dicritical digraphs.We finally study the notion of redicolouring for digraphs. In particular we prove that a large collection of evidences for Cereceda's conjecture admit a directed counterpart.