For a graph $G$, the $k$-recolouring graph $\mathcal{R}_k(G)$ is the graph whose vertices are the $k$-colourings of $G$ and two colourings are joined by an edge if they differ in colour on exactly one vertex. We prove that for all $n \ge 1$, there exists a $k$-colourable weakly chordal graph $G$ where $\mathcal{R}_{k+n}(G)$ is disconnected, answering an open question of Feghali and Fiala. We also show that for every $k$-colourable $3K_1$-free graph $G$, $\mathcal{R}_{k+1}(G)$ is connected with diameter at most $4|V(G)|$.

翻译：对于一个图形$G$, $k$- recolor 图形$\ mathcal{R ⁇ k(G)$(G) 是一个图表, 其顶点是$k$- 彩色, $G$和两个彩色, 如果颜色不同, 正好是一个顶点。 我们证明, 对于所有 $\ge 1 美元, 存在一个 $k$- 彩色的低色色色方块$G$, 其中$\ mathcal{R ⁇ k+n}(G) 断开, 回答Feghali 和 Fiala 的未决问题。 我们还显示, 每张可彩色的$3K_ 1美元无色方块$ G$, $\ mathcal{R ⁇ k+1} (G) 与直径有关, 最多为 $4V(G) 。