Given two $k$-dicolourings of a digraph $D$, we prove that it is PSPACE-complete to decide whether we can transform one into the other by recolouring one vertex at each step while maintaining a dicolouring at any step even for $k=2$ and for digraphs with maximum degree $5$ or oriented planar graphs with maximum degree $6$. A digraph is said to be $k$-mixing if there exists a transformation between any pair of $k$-colourings. We show that every digraph $D$ is $k$-mixing for all $k\geq \delta^*_{\min}(D)+2$, generalizing a result due to Dyer et al. We also prove that every oriented graph $\vec{G}$ is $k$-mixing for all $k\geq \delta^*_{\max}(\vec{G}) +1$ and for all $k\geq \delta^*_{\rm avg}(\vec{G})+1$. We conjecture that, for every digraph $D$, the dicolouring graph of $D$ on $k\geq \delta_{\min}^*(D)+2$ colours has diameter at most $O(|V(D)|^2)$ and give some evidences. We first prove that the dicolouring graph of any digraph $D$ on $k\geq 2\delta_{\min}^*(D) + 2$ colours has linear diameter, extending a result from Bousquet and Perarnau. We also prove that the conjecture is true when $k\geq \frac{3}{2}(\delta_{\min}^*(D)+1)$. Restricted to the special case of oriented graphs, we prove that the dicolouring graph of any subcubic oriented graph on $k\geq 2$ colours is connected and has diameter at most $2n$. We conjecture that every non $2$-mixing oriented graph has maximum average degree at least $4$, and we provide some support for this conjecture by proving it on the special case of $2$-freezable oriented graphs. More generally, we show that every $k$-freezable oriented graph on $n$ vertices must contain at least $kn + k(k-2)$ arcs, and we give a family of $k$-freezable oriented graphs that reach this bound. In the general case, we prove as a partial result that every non $2$-mixing oriented graph has maximum average degree at least $\frac{7}{2}$.

翻译：对于有向图$D$的两种$k$-dicolouring着色，我们证明：即便在$k=2$、最大度为5的有向图或最大度为6的定向平面图的情况下，判定能否通过每次重染一个顶点并始终维持dicolouring性质而将一种着色变换为另一种着色的问题是PSPACE完全的。若任意一对$k$-dicolouring着色之间存在变换，则称该有向图是$k$-混合的。我们证明：对所有满足$k\geq \delta^*_{\min}(D)+2$的$k$，任意有向图$D$都是$k$-混合的，这推广了Dyer等人的结果。我们还证明：对所有满足$k\geq \delta^*_{\max}(\vec{G})+1$和$k\geq \delta^*_{\rm avg}(\vec{G})+1$的$k$，任意定向图$\vec{G}$都是$k$-混合的。我们猜想：对于任意有向图$D$，其在$k\geq \delta_{\min}^*(D)+2$种颜色下的dicolouring图的直径不超过$O(|V(D)|^2)$，并给出部分证据。首先证明：任意有向图$D$在$k\geq 2\delta_{\min}^*(D)+2$种颜色下的dicolouring图具有线性直径，这扩展了Bousquet和Perarnau的结果。我们还证明当$k\geq \frac{3}{2}(\delta_{\min}^*(D)+1)$时该猜想成立。针对定向图的特例，我们证明任意次立方定向图在$k\geq 2$种颜色下的dicolouring图是连通的且直径不超过$2n$。我们猜想所有非$2$-混合定向图的最大平均度至少为$4$，并通过证明该猜想对$2$-可冻结定向图成立来提供支持。更一般地，我们证明任意$n$个顶点的$k$-可冻结定向图必须包含至少$kn + k(k-2)$条弧，并给出达到此界的$k$-可冻结定向图族。作为部分结果，我们证明在一般情形下所有非$2$-混合定向图的最大平均度至少为$\frac{7}{2}$。