The graph edge coloring problem is to color the edges of a graph such that adjacent edges receives different colors. Let G be a simple graph with maximum degree Δ. The minimum number of colors needed for such a coloring of G is called the chromatic index of G, written χ2˘7(G). We say G is of class one if χ2˘7(G)=Δ, otherwise it is of class 2. A majority of edge coloring papers is devoted to the Classification Problem for simple graphs. A graph G is said to be \emph{overfull} if ∣E(G)∣3˘eΔ⌊∣V(G)∣/2⌋. Hilton in 1985 conjectured that every graph G of class two with Δ(G)3˘e3∣V(G)∣ contains an overfull subgraph H with Δ(H)=Δ(G). In this thesis, I will introduce some of my researches toward the Classification Problem of simple graphs, and a new approach to the overfull conjecture together with some new techniques and ideas