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arXiv · 2610.31263

Kempe Reachability with $k\geq 3$ Colors Is $\mathsf{PSPACE}$-Complete on Claw-Free Graphs

Duc A. Hoang
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Abstract

A \emph{proper vertex coloring} assigns colors to vertices so that adjacent vertices have distinct colors. A \emph{Kempe change} in a proper vertex coloring exchanges two colors on one component of the subgraph induced by the vertices of those colors. For a fixed palette of $k$ colors, \textsc{Kempe Reachability} asks whether two given proper vertex colorings of the same graph are connected by a sequence of Kempe changes. This problem is known to be $\mathsf{PSPACE}$-complete for every fixed $k\geq 3$. A graph is \emph{claw-free} if no vertex has three pairwise nonadjacent neighbors. The \emph{line graph} of a graph has one vertex for each edge, with two vertices adjacent exactly when the corresponding edges share an end. Line graphs are claw-free, and their Kempe changes correspond to \emph{edge-Kempe changes} in the original graph, which exchange two colors on one component of the subgraph formed by the edges of those colors. A claw-free graph with a proper $k$-coloring has maximum degree at most $2k-2$, whereas for $k\geq 4$ all proper $k$-colorings of a graph of maximum degree at most $k$ are connected by Kempe changes. We show that, for every fixed $k\geq 3$, \textsc{Kempe Reachability} with $k$ colors is $\mathsf{PSPACE}$-complete even on line graphs of connected simple $k$-regular graphs, and hence even on connected $(2k-2)$-regular claw-free graphs. Equivalently, deciding whether two proper $k$-edge-colorings are connected by edge-Kempe changes is $\mathsf{PSPACE}$-complete even on connected simple $k$-regular graphs. ChatGPT 6 Pro and Claude Fable 5.1 were used to assist with developing and reviewing mathematical arguments and drafting and revising the paper. The author takes full responsibility for the correctness and content of the paper.

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Duc A. Hoang. 2026-10-22. Kempe Reachability with $k\geq 3$ Colors Is $\mathsf{PSPACE}$-Complete on Claw-Free Graphs. https://arxiv.org/abs/2610.31263

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