Local Reed Conjecture

Prove that every graph G is f(G)-colourable, where f(G) = max_{v ∈ V(G)} ⌈(ω(v) + deg(v) + 1)/2⌉ and ω(v) denotes the maximum size of a clique containing v.

Background

The local version strengthens Reed's conjecture by assigning each graph a colouring threshold based on the largest clique and degree in the neighbourhood of each individual vertex. The paper introduces this conjecture as an existing unresolved strengthening before discussing recolouring versions of Reed-type statements.

References

\begin{conjecture}[Local Reed Conjecture] All graph $G$ are $f(G)$-colourable. \end{conjecture}

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