The reconfiguration graph for the $k$-colourings of a graph $G$, denoted $R_{k}(G)$, 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. For any $k$-colourable $P_4$-free graph $G$, Bonamy and Bousquet proved that $R_{k+1}(G)$ is connected. In this short note, we complete the classification of the connectedness of $R_{k+1}(G)$ for a $k$-colourable graph $G$ excluding a fixed path, by constructing a $7$-chromatic $2K_2$-free (and hence $P_5$-free) graph admitting a frozen $8$-colouring. This settles a question of the second author.

翻译：以美元表示的G$美元(G)的彩色的重新配置图,是用美元彩色表示的G$(G)的图表,其顶点是美元彩色的G$和两个彩色的图,如果颜色不同,则加上一个边,只要一个顶点的颜色不同。对于任何可兑换美元彩色的P$4美元无G美元图,Bonamy和Bousquet证明,美元+1美元(G)是相连的。在本简短的说明中,我们完成了对美元彩色的G$(G)的关联性分类,其中不包括固定路径,我们用7美元的色素2美元(因此是P5美元免费的)图来表示冻结的8美元彩色。这解决了第二个作者的问题。