A Dichotomy Theorem for Circular Colouring Reconfiguration

TL;DR

This paper establishes a clear computational complexity dichotomy for the reconfiguration problem of circular graph colourings, showing polynomial-time solvability for certain ratios and PSPACE-completeness for others.

Contribution

It generalizes a known dichotomy theorem from classical to circular colourings, providing a comprehensive complexity classification.

Findings

Polynomial-time solvability for 2 ≤ p/q < 4

PSPACE-completeness for p/q ≥ 4

Generalization of classical colouring reconfiguration results

Abstract

The "reconfiguration problem" for circular colourings asks, given two $(p,q)$-colourings $f$ and $g$ of a graph $G$, is it possible to transform $f$ into $g$ by changing the colour of one vertex at a time such that every intermediate mapping is a $(p,q)$-colouring? We show that this problem can be solved in polynomial time for $2\leq p/q <4$ and is PSPACE-complete for $p/q\geq 4$. This generalizes a known dichotomy theorem for reconfiguring classical graph colourings.