Linear minimum-degree bound for bounded list lengths

Establish a minimum-degree threshold of order O(ℓε^{-1}) guaranteeing that every graph admits an α-majority edge colouring from every ε-excessive list assignment whose lists have size at most ℓ, for each fixed pair (ℓ,ε) with ℓ∈N and ε>0.

Background

The paper studies diversified majority tolerances, where each colour has its own tolerance, and defines ε-excessive list assignments by requiring the sum of tolerances in every list to exceed 1+ε. Theorem 16 gives a threshold of order O(ℓ2) under a bound on list lengths, while Theorem 19 obtains a stronger result in the special uniform-tolerance-vector setting.

Conjecture 23 asks whether the dependence on the maximum list length can be improved to linear order for arbitrary tolerance functions, without imposing a positive lower bound on individual tolerances.

References

Conjecture 23. There is a function 82 : N x (0, +00) 3 (l, 8) -> R such that d2 = O(le-1) and for any fixed (l, E) E Nx (0, +00) and every graph G with the minimum degree d(G) ≥ 82(l,E), any E-excessive list assignment L of G associated with arbitrary tolerance function a and containing exclusively list of lengths at most l, i.e. with |L(e)| < for every e E E(G), 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 23, Section 7 (Concluding remarks)