The Heawood map coloring conjecture [by] J. W. T. Youngs.
The four-color conjecture for a sphere is a famous unsolved problem, and the only information available today is that the chromatic number of a sphere is either four or five. On the other hand, it has been known for three-quarters of a century that the chromatic number of a torus is seven. This memorandum is concerned with the chromatic number of orientable two-manifolds of positive genus. The problem is not completely solved, but considerable progress has been made in the past few years. This study reports on current results and, above all, methods that have been employed.