a graph, when mentioned without any qualification, is always assumed to be a proper coloring of the edges, meaning no two adjacent edges are assigned the same color. Here, two distinct edges are considered to be adjacent when they share a common vertex. An edge coloring of a graph may also be thought of as equivalent to a vertex coloring of the line graph , the graph that has a vertex for every edge of and an edge for every pair of adjacent edges in .

A proper edge coloring with different colors