For a list-assignment $L$, the reconfiguration graph $C_L(G)$ of a graph $G$ is the graph whose vertices are proper $L$-colorings of $G$ and whose edges link two colorings that differ on only one vertex. If $|L(v)| \ge d(v) + 2$ for every vertex of $G$, it is known that $C_L(G)$ is connected. In this case, Cambie et al. investigated the diameter of $C_L(G)$. They conjectured that $diam(C_L(G)) \le n(G) + μ(G)$ with $μ(G)$ the size of a maximum matching of $G$ and proved several results towards this conjecture. We answer to two of their open problems by proving the conjecture for two classes of graphs, namely subcubic graphs and complete multipartite graphs.

翻译：对于列表赋值$L$，图$G$的重构图$C_L(G)$是以$G$的所有正常$L$着色为顶点、连接仅在一个顶点处不同的两个着色边的图。若对$G$的每个顶点$v$均有$|L(v)| \ge d(v) + 2$，则已知$C_L(G)$是连通的。此时，Cambie等人研究了$C_L(G)$的直径，并猜想$diam(C_L(G)) \le n(G) + μ(G)$，其中$μ(G)$为$G$的最大匹配大小，他们为验证该猜想证明了若干结果。我们通过证明该猜想对两类图（即次立方图与完全多部图）成立，回应了其两个开放问题。