Maximal k-edge-colorable subgraphs, Vizing's Theorem, and Tuza's Conjecture

We prove that if M is a maximal k-edge-colorable subgraph of a multigraph G and if F={v∈V(G):dM(v)≤k−μ(v)}, then dF(v)≤dM(v) for all v∈V(G) with dM(v)