To simplify, suppose that case k has been established if only a finite number of gs of the form 12s + k are in doubt.
Replacing the genus by the Euler characteristic, we obtain a formula that covers both the orientable and non-orientable cases, This relation holds, as Ringel and Youngs showed, for all surfaces except for the Klein bottle.
The Franklin graph can be drawn on the Klein bottle in a way that forms six mutually-adjacent regions, showing that this bound is tight.
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.