  Let $\chi'_\subset(G)$ be the least number of colours necessary to properly
colour the edges of a graph $G$ with minimum degree $\delta\geq 2$ so that the
set of colours incident with any vertex is not contained in a set of colours
incident to any its neighbour. We provide an infinite family of examples of
graphs $G$ with $\chi'_\subset(G)\geq (1+\frac{1}{\delta-1})\Delta$, where
$\Delta$ is the maximum degree of $G$, and we conjecture that
$\chi'_\subset(G)\leq \lceil(1+\frac{1}{\delta-1})\Delta\rceil$ for every
connected graph with $\delta\geq 2$ which is not isomorphic to $C_5$. The
equality here is attained e.g. for the family of complete bipartite graphs.
Using a probabilistic argument we support this conjecture by proving that for
any fixed $\delta\ge2$, $\chi'_\subset(G) \le (1+\frac{4}{\delta})\Delta
(1+o(1))$ (for $\Delta\to\infty$), what implies that $\chi'_\subset(G) \le
(1+\frac{4}{\delta-1})\Delta$ for $\Delta$ large enough.
