Recolouring odd-hole-free graphs at the Reed threshold

Prove that every odd-hole-free graph of maximum degree Δ and clique number ω is k-recolourable for all k ≥ ⌈(ω + Δ + 1)/2⌉.

Background

The paper considers the conjecture that odd-hole-free graphs satisfy a particularly strong Kempe-recolourability property at the Reed threshold. Earlier work established only weaker bounds, and the paper improves those results to within one colour. The authors explicitly identify the exact case k = f(G) as unresolved.

References

Therefore, the only remaining open case for \cref{conj:odd_hole_reed} is $k = f(G)$.

— A Recolouring Version of a Conjecture of Reed  (2502.10147 - Meyer et al., 14 Feb 2025) in Section 1, subsection “Recolouring odd-hole-free graphs”, Conjecture labelled \cref{conj:odd_hole_reed}