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 Δ max ( D ) ${{\rm{\Delta }}}_{\max }(D)$ as the maximum of { max ( d + ( v ) , d − ( v ) ) ∣ v ∈ V } $\{\max ({d}^{+}(v),{d}^{-}(v))| v\in V\}$ and Δ min ( D ) ${{\rm{\Delta }}}_{\min }(D)$ as the maximum of { min ( d + ( v ) , d − ( v ) ) ∣ v ∈ V } $\{\min ({d}^{+}(v),{d}^{-}(v))| v\in V\}$. It is known that the dichromatic number of D $D$ is at most Δ min ( D ) + 1 ${{\rm{\Delta }}}_{\min }(D)+1$. In this work, we prove that every digraph D $D$ which has dichromatic number exactly Δ min ( D ) + 1 ${{\rm{\Delta }}}_{\min }(D)+1$ must contain the directed join of K r ↔ $\overleftrightarrow{{K}_{r}}$ and K s ↔ $\overleftrightarrow{{K}_{s}}$ for some r , s $r,s$ such that r + s = Δ min ( D ) + 1 $r+s={{\rm{\Delta }}}_{\min }(D)+1$, except if Δ 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 Δ min ( G → ) ≥ 2 ${{\rm{\Delta }}}_{\min }(\overrightarrow{G})\ge 2$ has dichromatic number at most Δ min ( G → ) ${{\rm{\Delta }}}_{\min }(\overrightarrow{G})$. Let G → $\overrightarrow{G}$ be an oriented graph of order n $n$ such that Δ 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 Δ min ( G → ) n $2{{\rm{\Delta }}}_{\min }(\overrightarrow{G})n$ when k ≥ Δ 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 Δ max ( D ) = Δ ≥ 3 ${{\rm{\Delta }}}_{\max }(D)={\rm{\Delta }}\ge 3$ and every k ≥ Δ + 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 Δ n 2 ${c}_{{\rm{\Delta }}}{n}^{2}$, where c Δ = O ( Δ 2 ) ${c}_{{\rm{\Delta }}}=O({{\rm{\Delta }}}^{2})$ is a constant depending only on Δ ${\rm{\Delta }}$.