  A $\frac{1}{k}$-majority $l$-edge-colouring of a graph $G$ is a colouring of
its edges with $l$ colours such that for every colour $i$ and each vertex $v$
of $G$, at most $\frac{1}{k}$'th of the edges incident with $v$ have colour
$i$. We conjecture that for every integer $k\geq 2$, each graph with minimum
degree $\delta\geq k^2$ is $\frac{1}{k}$-majority $(k+1)$-edge-colourable and
observe that such result would be best possible. This was already known to hold
for $k=2$. We support the conjecture by proving it with $2k^2$ instead of
$k^2$, which confirms the right order of magnitude of the conjectured optimal
lower bound for $\delta$. We at the same time improve the previously known
bound of order $k^3\log k$, based on a straightforward probabilistic approach.
As this technique seems not applicable towards any further improvement, we use
a more direct non-random approach. We also strengthen our result, in particular
substituting $2k^2$ by $(\frac{7}{4}+o(1))k^2$. Finally, we provide the proof
of the conjecture itself for $k\leq 4$ and completely solve an analogous
problem for the family of bipartite graphs.
