Coloring Reconfiguration under Color Swapping

TL;DR

The paper introduces color swapping (CS) as a novel reconfiguration rule for graph colorings and studies the corresponding problems CRCS and k-CRCS. It establishes a dichotomy in the number of colors: CRCS is solvable in polynomial time for $k\le 2$ and is $ ext{PSPACE}$-complete for $k\ge 3$, with hardness on bipartite, split, and chordal graphs under appropriate regimes. It provides polynomial-time algorithms for several graph classes, including 3-CRCS on paths and fixed-$k$ split graphs, and for cographs, while delivering sweeping hardness results via reductions from Token Sliding, Coloring Reconfiguration, and NCL to map the reconfiguration landscape onto restricted graph families. The work also ties CS to Kawasaki dynamics in physics and positions CS as a restricted variant of Kempe-chain recoloring, enriching the broader study of reconfiguration problems and their computational boundaries. Together, these results illuminate the complexity landscape of CS-based color reconfiguration and offer practical algorithms for structurally restricted instances.

Abstract

In the \textsc{Coloring Reconfiguration} problem, we are given two proper $k$-colorings of a graph and asked to decide whether one can be transformed into the other by repeatedly applying a specified recoloring rule, while maintaining a proper coloring throughout. For this problem, two recoloring rules have been widely studied: \emph{single-vertex recoloring} and \emph{Kempe chain recoloring}. In this paper, we introduce a new rule, called \emph{color swapping}, where two adjacent vertices may exchange their colors, so that the resulting coloring remains proper, and study the computational complexity of the problem under this rule. We first establish a complexity dichotomy with respect to $k$: the problem is solvable in polynomial time for $k \leq 2$, and is PSPACE-complete for $k \geq 3$. We further show that the problem remains PSPACE-complete even on restricted graph classes, including bipartite graphs, split graphs, and planar graphs of bounded degree. In contrast, we present polynomial-time algorithms for several graph classes: for paths when $k = 3$, for split graphs when $k$ is fixed, and for cographs when $k$ is arbitrary.

Figures (7)

• Figure 1: A reconfiguration sequence between two proper $3$-colorings $f_s$ and $f_t$ under color swapping.
• Figure 2: Our results for graph classes. Each arrow represents the inclusion relationship between classes: $A\rightarrow B$ means that the graph class $B$ is a proper subclass of the graph class $A$.
• Figure 4: Construction of $G$ from $G'$ using four colors. Vertices of $G$ are assigned a proper $4$-coloring $g_t$, and those of $G'$ the corresponding proper $4$-coloring $f_t$.
• Figure 5: (a) A configuration of an NCL machine, (b) an and vertex, and (c) an or vertex. Edges of weight $2$ are shown in blue lines, and edges of weight $1$ in red lines. The NCL machine in (a) is an and/or constraint graph.
• Figure 6: (a) An illustration of a 3-forbidden pendant for a vertex $x$, which ensures that $x$ is never assigned color $3$. (b) A simplified depiction of the gadget used to represent this pendant.

Theorems & Definitions (6)

• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 7
• Theorem 8