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

Let D=(V,A) $D=(V,A)$ be a digraph. We define Delta max(D) ${{\rm{\Delta }}}_{\max }(D)$ as the maximum of {max(d+(v),d-(v)) divide v is an element of V} $\{\max ({d}<^>{+}(v),{d}<^>{-}(v))| v\in V\}$ and Delta min(D) ${{\rm{\Delta }}}_{\min }(D)$ as the maximum of {min(d+(v),d-(v)) divide v is an element of V} $\{\min ({d}<^>{+}(v),{d}<^>{-}(v))| v\in V\}$. It is known that the dichromatic number of D $D$ is at most Delta min(D)+1 ${{\rm{\Delta }}}_{\min }(D)+1$. In this work, we prove that every digraph D $D$ which has dichromatic number exactly Delta min(D)+1 ${{\rm{\Delta }}}_{\min }(D)+1$ must contain the directed join of Kr <-> $\overleftrightarrow{{K}_{r}}$ and Ks <-> $\overleftrightarrow{{K}_{s}}$ for some r,s $r,s$ such that r+s=Delta min(D)+1 $r+s={{\rm{\Delta }}}_{\min }(D)+1$, except if Delta min(D)=2 ${{\rm{\Delta }}}_{\min }(D)=2$ in which case D $D$ must contain a digon. In particular, every oriented graph G -> $\overrightarrow{G}$ with Delta min(G ->)>= 2 ${{\rm{\Delta }}}_{\min }(\overrightarrow{G})\ge 2$ has dichromatic number at most Delta min(G ->) ${{\rm{\Delta }}}_{\min }(\overrightarrow{G})$. Let G -> $\overrightarrow{G}$ be an oriented graph of order n $n$ such that Delta min(G ->)<= 1 ${{\rm{\Delta }}}_{\min }(\overrightarrow{G})\le 1$. Given two 2-dicolourings of G -> $\overrightarrow{G}$, we show that we can transform one into the other in at most n $n$ steps, by recolouring exactly one vertex at each step while maintaining a dicolouring at any step. Furthermore, we prove that, for every oriented graph G -> $\overrightarrow{G}$ on n $n$ vertices, the distance between two k $k$-dicolourings is at most 2 Delta min(G ->)n $2{{\rm{\Delta }}}_{\min }(\overrightarrow{G})n$ when k >=Delta min(G ->)+1 $k\ge {{\rm{\Delta }}}_{\min }(\overrightarrow{G})+1$. We then extend a theorem of Feghali, Johnson and Paulusma to digraphs. We prove that, for every digraph D $D$ with Delta max(D)=Delta >= 3 ${{\rm{\Delta }}}_{\max }(D)={\rm{\Delta }}\ge 3$ and every k >=Delta+1 $k\ge {\rm{\Delta }}+1$, the k $k$-dicolouring graph of D $D$ consists of isolated vertices and at most one further component that has diameter at most c Delta n2 ${c}_{{\rm{\Delta }}}{n}<^>{2}$, where c Delta=O(Delta 2) ${c}_{{\rm{\Delta }}}=O({{\rm{\Delta }}}<^>{2})$ is a constant depending only on Delta ${\rm{\Delta }}$.