Abstract: Vizing’s Theorem states that every graph of maximum degree \(\Delta\) admits a proper edge-coloring using at most \(\Delta + 1\) colors. There has been a flurry of recent works on fast edge coloring algorithms. In this talk, I will discuss classical and more recent approaches toward near-Vizing edge-coloring algorithms, i.e., \((1+\epsilon)\Delta\)-edge-coloring. The focus will be on the so-called augmenting subgraph approach.