In this paper we introduce and study a new problem named min-max edge q -coloring which is motivated by applications in wireless mesh networks. The input of the problem consists of an undirected graph and an integer q. The goal is to color the edges of the graph with as many colors as possible such that: (a) any vertex is incident to at most q different colors, and (b) the maximum size of a color group (i.e. set of edges identically colored) is minimized. We show the following results: 1. Min-max edge q-coloring is NP-hard, for any q>=2. 2. A polynomial time exact algorithm for min-max edge q-coloring on trees. 3. Exact formulas of the optimal solution for cliques. 4. An approximation algorithm for planar graphs.