Algorithms
Because the problem of testing whether a graph is class 1 is NP-complete, there is no known polynomial time algorithm for edge-coloring every graph with an optimal number of colors. Nevertheless a number of algorithms have been developed that relax one or more of these criteria: they only work on a subset of graphs, or they do not always use an optimal number of colors, or they do not always run in polynomial time.
Read more about this topic: Edge Coloring