Coloring Reconfiguration under Color Swapping

TL;DR

This paper introduces a new color swapping rule for graph coloring reconfiguration, establishing complexity results and polynomial algorithms across various graph classes, advancing understanding of coloring transformations.

Contribution

It defines the color swapping reconfiguration problem, provides a complexity dichotomy based on the number of colors, and offers polynomial algorithms for specific graph classes.

Findings

Problem is polynomial-time solvable for k ≤ 2.

Problem is PSPACE-complete for k ≥ 3.

Polynomial algorithms exist for paths, split graphs, and cographs.

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.