  In the last years, connection concepts such as rainbow connection and proper
connection appeared in graph theory and obtained a lot of attention. In this
paper, we investigate the loose edge-connection of graphs. A connected
edge-coloured graph $G$ is loose edge-connected if between any two of its
vertices there is a path of length one, or a bi-coloured path of length two, or
a path of length at least three with at least three colours used on its edges.
The minimum number of colours, used in a loose edge-colouring of $G$, is called
the loose edge-connection number and denoted $\lec(G)$. We determine the
precise value of this parameter for any simple graph $G$ of diameter at least
3. We show that deciding, whether $\lec(G) = 2$ for graphs $G$ of diameter 2,
is an NP-complete problem. Furthermore, we characterize all complete bipartite
graphs $K_{r,s}$ with $\lec(K_{r,s}) = 2$.
