Maximal kk-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 $F = \{v \in V(G) : d_M(v) \leq k-\mu(v)\}$, then $d_F(v) \leq d_M(v)$ for all $v \in F$. (When $G$ is a simple graph, the set $F$ is just the set of vertices having degree less than $k$ in $M$.) This implies Vizing's Theorem as well as a special case of Tuza's Conjecture on packing and covering of triangles. A more detailed version of our result also implies Vizing's Adjacency Lemma for simple graphs.Comment: 11 pages, 1 figure. Fixed some inaccurate references to "Vizing's Theorem" (the stronger version cited here is in fact due to Ore), cleared up some muddled results in the section about forests, simplified some notation, and made other various readability improvement