Complexity of graph isomorphism with properly colored edges (ref. request)
In general, if the number of adjacent edges, which have the same color, is bounded by a constant, say d. Then, the isomorphism problem for n-vertex graphs can be solved in n^(cd) for some constant c. In your case, d=1. See for example, Proposition 4.5 in CANONICAL LABELING OF GRAPHS by Babai and Luks. Actually, they considered vertex coloring, but the same arguments works for edge colors as well. For example, you could subdivide each edge and instert a new vertex that encodes the edge color.