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.
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