Edge coloring of a graph is an assignment of “colors” to the edges of the graph so that no two adjacent edges have the same color with an optimal number of colors. Two edges are said to be adjacent if they are connected to the same vertex.

There is no known polynomial time algorithm for edge-coloring every graph with an optimal number of colors. Nevertheless, is there any efficient algorithm for edge-coloring every complete graph with an optimal number of colors? I bielieve this exits but I cannot find any source for this.