Heuristic algorithms for coloring the edges of large undirected single-edge graphs with (or very close to) the minimal number of colors are presented. Compared to simulated annealing and a grouping genetic algorithm for small graphs, the heuristics were not only faster by orders of magnitude, but almost all solutions had the optimal color number; the rest differed by at most two colors. For large graphs, the heuristics were validated by an evolutionary algorithm. Here, the heuristics often found an optimum or a solution very close to it.