General majority edge-colouring threshold

Prove that every graph with minimum degree at least k^2 admits a 1/k-majority edge colouring with k+1 colours for every integer k≥2.

Background

The paper recalls a conjecture from the non-list setting asserting that the minimum-degree threshold k2 should suffice for a 1/k-majority edge colouring with k+1 colours. The conjecture is tight up to the boundary because the paper cites a graph of minimum degree k2−1 that does not admit such a colouring.

The conjecture is known for k=2,3,4 and for bipartite graphs, but it remains unresolved in general. Later results in the paper establish further special cases, including regular graphs, without settling the full conjecture.

References

It was also conjectured that the following holds true. Note that in view of Observation 3, we cannot expect anything more.

Conjecture 5 ([19]). For every integer k ≥ 2, if a graph G has minimum degree 8 ≥ k2, then G is -- majority edge (k + 1)-colourable.

— On list extensions of the majority edge colourings  (2502.12688 - Pękała et al., 18 Feb 2025) in Conjecture 5, Section 1