Recolouring version of Reed's conjecture for general graphs

Prove that every graph G is k-recolourable for all k ≥ ⌈(1/3)ω(G) + (2/3)(Δ(G)+1)⌉.

Background

The conjecture proposes the sharp threshold suggested by the paper's analysis of frozen colourings. A construction gives non-recolourable colourings above the 1/3 threshold, while the paper proves that non-unique frozen colourings cannot occur at or below the corresponding threshold. Full recolourability would require addressing obstructions beyond frozen colourings.

References

\begin{conjecture}\label{conj:recol_reed} Any graph $G$ is $k$-recolourable for all $k \lceil \frac{1}{3}\omega(G) + {\frac{2}{3}(\Delta(G) +1)} \rceil$. \end{conjecture}

— A Recolouring Version of a Conjecture of Reed  (2502.10147 - Meyer et al., 14 Feb 2025) in Section 1, subsection “A recolouring version of Reed's conjecture”, Conjecture labelled \cref{conj:recol_reed}