Approximation and Hardness Results for the Maximum Edge q-coloring Problem

We consider the problem of coloring edges of a graph subject to the following constraint: for every vertex v, all the edges incident to v have to be colored with at most q colors. The goal is to find a coloring satisfying the above constraint and using the maximum number of colors. This problem has been studied in the past from the combinatorial and algorithmic point of view. The optimal coloring is known for some special classes of graphs. There is also an approximation algorithm for general graphs, which in the case q = 2 gives a 2-approximation. However, the complexity of finding the optimal coloring was not known. We prove that for any integer q >= 2 the problem is NP-Hard and APX-Hard. We also present a 5/3-approximation algorithm for q = 2 for graphs with a perfect matching.