Improved non-list threshold under a positive minimum tolerance

Establish a minimum-degree threshold of order O(α^{-1}ε^{-1}) guaranteeing an α-majority edge colouring for every ε-excessive tolerance function whose minimum tolerance is at least α.

Background

Conjecture 26 is the non-list analogue of Conjecture 24. It asks for a sharper degree threshold when every colour tolerance is bounded below by α.

The conjectured order improves the paper's currently available dependence, which is of order O(α{-1}ε{-2} log(αε){-1}). The paper explicitly states that proving this conjecture may require new tools and approaches.

References

Conjecture 26. There is a function 85 : (0, 1) x (0, +00) } (a, €) -> R such that ds = O(a-18-1) and for any fixed (a, E) € (0,1) x (0, +00) and every graph G with the minimum degree d(G) ≥ 85(a, E) and any E-excessive tolerance function a : C -> (0,1) with min(a) ≥ a, there is an a-majority C-colouring of G.

— On list extensions of the majority edge colourings  (2502.12688 - Pękała et al., 18 Feb 2025) in Conjecture 26, Section 7 (Concluding remarks)