Distance Recoloring

TL;DR

This work studies the reconfiguration complexity of $(d,k)$-colorings, where adjacent vertices within distance $\le d$ must differ in color, and each color assignment yields a feasible recoloring sequence under a one-vertex-at-a-time rule. The authors prove PSPACE-hardness for $d\ge 2$ and $k=\Omega(d^2)$ even on bipartite planar and $2$-degenerate graphs, via a two-layer reduction chain that leverages frozen graphs and forbidding paths, along with a Sliding Tokens variant; they also obtain a split-graph dichotomy and extend hardness to chordal graphs for even $d$, while giving quadratic-time algorithms for paths and efficient results for diameter-bounded graphs. The key technical contributions include a reduction framework from List $(d,k)$-Coloring Reconfiguration to $(d,k')$-Coloring Reconfiguration, a carefully designed Sliding Tokens reduction, and a color-count reduction to $O(d^2)$ colors, enabling hardness results on restricted graph classes. Together, these results delineate the hardness landscape of distance recoloring and provide efficient algorithms on specific structured graphs, highlighting practical implications for reconfiguration in constrained coloring problems.

Abstract

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 $α$ of a graph $G$ to reach another $(d, k)$-coloring $β$ 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 $k = Ω(d^2)$ 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 $k \geq d+1$.

Figures (8)

• Figure 1: An example of a vertex $v$ joining to its corresponding frozen graph $F_v$. Here $d$ is even, $k = 5$, $G$ is some list $(d,k)$-colorable graph having six vertices labelled $v, u_1, u_2, \dots, u_5$ (note that $u = u_i$ for some $i \in \{1, \dots, 5\}$), and $L(v) = \{2, 3\} \subseteq \{1, \dots, 5\}$.
• Figure 2: Rule \ref{['item:rule:deg4']} applied to a link edge of degree $4$
• Figure 3: Rule \ref{['item:rule:2deg3']} applied to a link edge joining two degree $3$ token triangles
• Figure 4: Rule \ref{['item:rule:2deg3']} applied to a link edge joining two degree $3$ token edges
• Figure 5: Rule \ref{['item:rule:2deg3']} applied to a link edge joining a degree $3$ token edge and a degree $3$ token triangle

Theorems & Definitions (55)

• theorem 1: Main Result
• theorem 2
• theorem 3
• theorem 4
• theorem 5
• theorem 5: Main Result
• theorem 6
• lemma 7
• proof
• lemma 8