The reconfiguration graph $\mathcal{C}_k(G)$ for the $k$-colourings of a graph $G$ has a vertex for each proper $k$-colouring of $G$, and two vertices of $\mathcal{C}_k(G)$ are adjacent precisely when those $k$-colourings differ on a single vertex of $G$. Much work has focused on bounding the maximum value of ${\rm{diam}}~\mathcal{C}_k(G)$ over all $n$-vertex graphs $G$. We consider the analogous problems for list colourings and for correspondence colourings. We conjecture that if $L$ is a list-assignment for a graph $G$ with $|L(v)|\ge d(v)+2$ for all $v\in V(G)$, then ${\rm{diam}}~\mathcal{C}_L(G)\le n(G)+\mu(G)$. We also conjecture that if $(L,H)$ is a correspondence cover for a graph $G$ with $|L(v)|\ge d(v)+2$ for all $v\in V(G)$, then ${\rm{diam}}~\mathcal{C}_{(L,H)}(G)\le n(G)+\tau(G)$. (Here $\mu(G)$ and $\tau(G)$ denote the matching number and vertex cover number of $G$.) For every graph $G$, we give constructions showing that both conjectures are best possible. Our first main result proves the upper bounds (for the list and correspondence versions, respectively) ${\rm{diam}}~\mathcal{C}_L(G)\le n(G)+2\mu(G)$ and ${\rm{diam}}~\mathcal{C}_{(L,H)}(G)\le n(G)+2\tau(G)$. Our second main result proves that both conjectured bounds hold, whenever all $v$ satisfy $|L(v)|\ge 2d(v)+1$. We conclude by proving one or both conjectures for various classes of graphs such as complete bipartite graphs, subcubic graphs, cactuses, and graphs with bounded maximum average degree.

翻译：重构图$\mathcal{C}_k(G)$对应于图$G$的$k$-染色，其顶点为$G$的所有合法$k$-染色，且$\mathcal{C}_k(G)$中两个顶点相邻当且仅当这两个$k$-染色在$G$的单个顶点上取值不同。现有研究主要聚焦于在所有$n$顶点图$G$上界定${\rm{diam}}~\mathcal{C}_k(G)$的最大值。本文考虑列表染色和对应染色的类似问题。我们猜想：若$L$是图$G$的一个列表赋值，且对所有$v\in V(G)$满足$|L(v)|\ge d(v)+2$，则${\rm{diam}}~\mathcal{C}_L(G)\le n(G)+\mu(G)$。同时猜想：若$(L,H)$是图$G$的一个对应覆盖，且对所有$v\in V(G)$满足$|L(v)|\ge d(v)+2$，则${\rm{diam}}~\mathcal{C}_{(L,H)}(G)\le n(G)+\tau(G)$。（此处$\mu(G)$和$\tau(G)$分别表示$G$的匹配数和顶点覆盖数。）对任意图$G$，我们构造了实例证明这两个猜想均为最优界。我们的第一个主要结果证明了上界${\rm{diam}}~\mathcal{C}_L(G)\le n(G)+2\mu(G)$和${\rm{diam}}~\mathcal{C}_{(L,H)}(G)\le n(G)+2\tau(G)$（分别对应列表版本和对应版本）。第二个主要结果证明：当所有顶点$v$满足$|L(v)|\ge 2d(v)+1$时，两个猜想界均成立。最后，我们针对完全二部图、次三次图、仙人掌图及有界最大平均度图等图类，证明了其中一个或两个猜想。