Improved threshold under a positive minimum tolerance

Establish a minimum-degree threshold of order O(α^{-1}ε^{-1}) guaranteeing that every graph admits an α-majority edge colouring from every ε-excessive list assignment associated with a tolerance function whose minimum tolerance is at least α.

Background

For tolerance functions bounded below by α, Theorem 15 provides a threshold of order O(α{-1}ε{-2} log(αε){-1}). The conjecture proposes improving the dependence on ε and the logarithmic factor to obtain a threshold of order O(α{-1}ε{-1}).

The statement concerns arbitrary list assignments satisfying the ε-excess condition and is therefore broader than the uniform tolerance-vector setting treated separately in the paper.

References

Conjecture 24. There is a function 83 : (0, 1) x (0, +00) } (a, €) -> R such that 83 = O(a-16-1) and for any fixed (a, E) € (0,1) x (0, +00) and every graph G with the minimum degree d(G) ≥ 83(a, E), any &- excessive list assignment L of G and every tolerance function a of L with min(a) ≥ a, there is an a-majority L-colouring of G.

— On list extensions of the majority edge colourings  (2502.12688 - Pękała et al., 18 Feb 2025) in Conjecture 24, Section 7 (Concluding remarks)