Extending an edge‐coloring

When can a k‐edge‐coloring of a subgraph K of a graph G be extended to a k‐edge‐coloring of G? One necessary condition is that (Formula Presented.) for all X ⊆ E(G) ‐ E(K), where μ_i(X) is the maximum cardinality of a subset of X whose union with the set of edges of K colored i is a matching. This condition is not sufficient in general, but is sufficient for graphs of very simple structure. We try to locate the border where sufficiency ends.