Strategies for Multigraph Edge Coloring

n mathematical graph theory, coloring problems are ubiquitous. Dating back to the famous four-color map problem, both the theory and applications associated with graph coloring have a rich history. A standard problem in graph theory is to color a graph’s vertices with the fewest number of colors such that no two adjacent vertices have the same color. This article examines a complementary problem with important applications in communications and scheduling. In particular, it considers the problem of optimally coloring a multigraph’s edges (where multiple edges are permitted between vertices) such that no edges incident at a given vertex share the same color. The article explores the theory surrounding the problem and surveys some algorithms that give nearly optimal solutions. Although powerful, these algorithms have important theoretical limitations. These limitations are discussed and recent conceptual and algorithmic techniques that appear to be promising candidates for circumventing the difficulties are reviewed.