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Edge coloring of a graph refers to the assignment of colors to the edges so that no two adjacent edges have the same color. The minimum number of colors needed is known as the chromatic index or edge chromatic number. Methods used include Vizing's Theorem and the Tait Coloring for cubic graphs.

In graph theory, a matching is a set of edges without common vertices. Methods to achieve maximum matchings include augmenting path algorithms and the Edmonds' Blossom Algorithm for general graphs.

An edge cover of a graph is a set of edges such that every vertex of the graph is incident to at least one edge of the set. The goal is to find the minimum edge cover. Methods include reduction to matching problems and using the Hungarian Method for minimum weight edge cover in weighted graphs.

Vizing's Theorem states that for a simple graph with maximum degree $\Delta$, the chromatic index is either $\Delta$ or $\Delta + 1$. It helps in edge coloring by providing a tight bound on the number of colors needed.

Tait Coloring is specific to cubic graphs which are 3-regular graphs. It states that edge-coloring a cubic graph with 3 colors is equivalent to finding a Hamiltonian decomposition of the graph. Tait's conjecture was that every 3-connected, planar cubic graph is 3-edge-colorable, but this was disproven by counterexamples.