Reed's chromatic number conjecture

Prove that every graph G satisfies χ(G) ≤ ⌈(ω(G) + Δ(G) + 1)/2⌉, thereby establishing Reed's conjectured bound relating chromatic number, clique number, and maximum degree.

Background

The paper begins by recalling Reed's conjecture, which asserts that the chromatic number of a graph is bounded by the average of its clique-number bound and maximum-degree bound. The conjecture is presented as a central motivation for studying recolouring analogues, and the paper notes that only weaker bounds are currently known in general.

References

Reed conjectured that the chromatic number of any graph~$G$ is at most~$\left\lceil \left(\omega(G) + \Delta(G) +1\right) / 2\right\rceil$.

— A Recolouring Version of a Conjecture of Reed  (2502.10147 - Meyer et al., 14 Feb 2025) in Section 1, Introduction