  Following a given ordering of the edges of a graph $G$, the greedy edge
colouring procedure assigns to each edge the smallest available colour. The
minimum number of colours thus involved is the chromatic index $\chi'(G)$, and
the maximum is the so-called Grundy chromatic index. Here, we are interested in
the restricted case where the ordering of the edges builds the graph in a
connected fashion. Let $\chi_c'(G)$ be the minimum number of colours involved
following such an ordering. We show that it is NP-hard to determine whether
$\chi_c'(G)>\chi'(G)$. We prove that $\chi'(G)=\chi_c'(G)$ if $G$ is bipartite,
and that $\chi_c'(G)\leq 4$ if $G$ is subcubic.
