Edge coloring

In graph theory, a proper edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring. The edge-coloring problem asks whether it is possible to color the edges of a given graph using at most k different colors, for a given value of k, or with the fewest possible colors. The minimum required number of colors for the edges of a given graph is called the chromatic index of the graph. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three. By Vizing's theorem, the number of colors needed to edge color a simple graph is either its maximum degree Δ or Δ+1. For some graphs, such as bipartite graphs and high-degree planar graphs, the number of colors is always Δ, and for multigraphs, the number of colors may be as large as 3Δ/2. There are polynomial time algorithms that construct optimal colorings of bipartite graphs, and colorings of non-bipartite simple graphs that use at most Δ+1 colors; however, the general problem of finding an optimal edge coloring is NP-hard and the fastest known algorithms for it take exponential time. Many variations of the edge-coloring problem, in which an assignments of colors to edges must satisfy other conditions than non-adjacency, have been studied. Edge colorings have applications in scheduling problems and in frequency assignment for fiber optic networks. A cycle graph may have its edges colored with two colors if the length of the cycle is even: simply alternate the two colors around the cycle. However, if the length is odd, three colors are needed. A complete graph Kn with n vertices is edge-colorable with n − 1 colors when n is an even number; this is a special case of Baranyai's theorem.