Let G be a (simple) graph with maximum degree three and chromatic index four. A 3-edge-coloring of G is a coloring of its edges in which only three colors are used. Then a vertex is conflicting when some edges incident to it have the same color. The minimum possible number of conflicting vertices that a 3- edge-coloring of G can have, d(G), is called the edge-coloring degree of G. Here we are mainly interested in the structure of a graph G with given edge-coloring degree and, in particular, when G is c-critical, that is d(G) = c = 1 and d(G - e) < c for any edge e of G.

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