Distance Recoloring

For integers $d \geq 1$ and $k \geq d+1$, the \textsc{Distance Coloring} problem asks if a given graph $G$ has a $(d, k)$-coloring, i.e., a coloring of the vertices of $G$ by $k$ colors such that any two vertices within distance $d$ from each other have different colors. In particular, the well-known \textsc{Coloring} problem is a special case of \textsc{Distance Coloring} when $d = 1$. For integers $d \geq 2$ and $k \geq d+1$, the \textsc{$(d, k)$-Coloring Reconfiguration} problem asks if there is a way to change the color of one vertex at a time, starting from a $(d, k)$-coloring $\alpha$ of a graph $G$ to reach another $(d, k)$-coloring $\beta$ of $G$, such that all intermediate colorings are also $(d, k)$-colorings. We show that even for planar, bipartite, and $2$-degenerate graphs, \textsc{$(d, k)$-Coloring Reconfiguration} remains $\mathsf{PSPACE}$-complete for $d \geq 2$ and $k = \Omega(d^2)$ via a reduction from the well-known \textsc{Sliding Tokens} problem. Additionally, on split graphs, there is an interesting dichotomy: the problem is $\mathsf{PSPACE}$-complete when $d = 2$ and $k$ is large but can be solved efficiently when $d \geq 3$ and $k \geq d+1$. For chordal graphs, we show that the problem is $\mathsf{PSPACE}$-complete for even values of $d \geq 2$. Finally, we design a quadratic-time algorithm to solve the problem on paths for any $d \geq 2$ and $k \geq d+1$.