Strengthening the directed Brooks' theorem for oriented graphs and consequences on digraph redicolouring

Abstract

AbstractLet be a digraph. We define as the maximum of and as the maximum of . It is known that the dichromatic number of is at most . In this work, we prove that every digraph which has dichromatic number exactly must contain the directed join of and for some such that , except if in which case must contain a digon. In particular, every oriented graph with has dichromatic number at most . Let be an oriented graph of order such that . Given two 2‐dicolourings of , we show that we can transform one into the other in at most steps, by recolouring exactly one vertex at each step while maintaining a dicolouring at any step. Furthermore, we prove that, for every oriented graph on vertices, the distance between two ‐dicolourings is at most when . We then extend a theorem of Feghali, Johnson and Paulusma to digraphs. We prove that, for every digraph with and every , the ‐dicolouring graph of consists of isolated vertices and at most one further component that has diameter at most , where is a constant depending only on .