The upper bound, proved in Heawood's original short paper, is straightforward: by manipulating the Euler characteristic, one can show that any graph embedded in the surface must have at least one vertex of degree less than the given bound.
The upper bound, proved in Heawood's original short paper, is based on a greedy coloring algorithm. By manipulating the Euler characteristic, one can show that every graph embedded in the given surface must have at least one vertex of degree less than the given bound.