Strengthening the Directed Brooks' Theorem for oriented graphs and consequences on digraph redicolouring

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