Given a digraph $D=(V,A)$ on $n$ vertices and a vertex $v\in V$, the cycle-degree of $v$ is the minimum size of a set $S \subseteq V(D) \setminus \{v\}$ intersecting every directed cycle of $D$ containing $v$. From this definition of cycle-degree, we define the $c$-degeneracy (or cycle-degeneracy) of $D$, which we denote by $\delta^*_c(D)$. It appears to be a nice generalisation of the undirected degeneracy. In this work, using this new definition of cycle-degeneracy, we extend several evidences for Cereceda's conjecture to digraphs. The $k$-dicolouring graph of $D$, denoted by $\mathcal{D}_k(D)$, is the undirected graph whose vertices are the $k$-dicolourings of $D$ and in which two $k$-dicolourings are adjacent if they differ on the colour of exactly one vertex. We show that $\mathcal{D}_k(D)$ has diameter at most $O_{\delta^*_c(D)}(n^{\delta^*_c(D) + 1})$ (respectively $O(n^2)$ and $(\delta^*_c(D)+1)n$) when $k$ is at least $\delta^*_c(D)+2$ (respectively $\frac{3}{2}(\delta^*_c(D)+1)$ and $2(\delta^*_c(D)+1)$). This improves known results on digraph redicolouring (Bousquet et al.). Next, we extend a result due to Feghali to digraphs, showing that $\mathcal{D}_{d+1}(D)$ has diameter at most $O_{d,\epsilon}(n(\log n)^{d-1})$ when $D$ has maximum average cycle-degree at most $d-\epsilon$. We then show that two proofs of Bonamy and Bousquet for undirected graphs can be extended to digraphs. The first one uses the digrundy number of a digraph and the second one uses the $\mathscr{D}$-width. Finally, we give a general theorem which makes a connection between the recolourability of a digraph $D$ and the recolourability of its underlying graph $UG(D)$. This result directly extends a number of results on planar graph recolouring to planar digraph redicolouring.

翻译：给定$n$个顶点的有向图$D=(V,A)$及顶点$v\in V$，$v$的圈度是包含$v$的所有有向环中，与$V(D) \setminus \{v\}$交集的最小集合$S$的大小。基于圈度的定义，我们定义$D$的$c$-退化度（或称圈退化度），记为$\delta^*_c(D)$。该概念似乎是对无向退化度的良好推广。本文利用圈退化度这一新定义，将Cereceda猜想的若干证据拓展至有向图。$D$的$k$-着色图$\mathcal{D}_k(D)$是无向图，其顶点为$D$的$k$-着色，两个$k$-着色相邻当且仅当它们恰有一个顶点的颜色不同。我们证明：当$k$至少为$\delta^*_c(D)+2$（分别为$\frac{3}{2}(\delta^*_c(D)+1)$和$2(\delta^*_c(D)+1)$）时，$\mathcal{D}_k(D)$的直径不超过$O_{\delta^*_c(D)}(n^{\delta^*_c(D) + 1})$（分别为$O(n^2)$和$(\delta^*_c(D)+1)n$）。这改进了Bousquet等人关于有向图重着色的已知结果。其次，我们将Feghali的结果推广至有向图：当$D$的最大平均圈度不超过$d-\epsilon$时，$\mathcal{D}_{d+1}(D)$的直径至多为$O_{d,\epsilon}(n(\log n)^{d-1})$。随后证明Bonamy和Bousquet关于无向图的两个证明可推广至有向图：第一个利用有向图的基度数，第二个利用$\mathscr{D}$-宽度。最后，我们给出一个通用定理，建立了有向图$D$的可重着色性与其底层图$UG(D)$的可重着色性之间的联系。该结果直接扩展了若干平面图重着色结论至平面有向图重着色领域。